Analysis Problem Book Amol Sasane

Analysis Problem BookAmol Sasane

ivContentsChapter 5. Integration 27§5.1. Definition and properties of the Riemann integral 27§5.2. Fundamental Theorem of Integral Calculus 27§5.3. Integration by Parts and by Substitution 28§5.4. Riemann sums 29§5.5. Improper integrals 29§5.6. Transcendental functions 32§5.7. Applications: length, area, volume 37Chapter 6. Series 41§6.1. Convergence/divergence of series 41§6.2. Absolute convergence and the Leibniz’s Alternating Series Test 42§6.3. Comparison, Ratio, Root 42§6.4. Power series 45Solutions 47Solutions to the exercises from Chapter 1 47Solutions to the exercises from Chapter 2 68Solutions to the exercises from Chapter 3 82Solutions to the exercises from Chapter 4 94Solutions to the exercises from Chapter 5 118Solutions to the exercises from Chapter 6 154

Chapter 1The real number systemand preliminaries1.1. Sum of squares1.1.1. The aim of this exercise is to offer a simple justification for the sum of squares formula.(1) Show using induction that 1 2 +2 2 +3 2 +···+n 2 = n(n+1)(2n+1) for all n ∈ N.6(2) Here is a pictorial “proof without words” of the fact that for n ∈ N,⎧•⎪⎨• •n• • • ⎪⎩• • • •1+2+3+···+n = n(n+1) .2+◦ ◦ ◦ ◦◦ ◦ ◦◦ ◦◦=• ◦ ◦ ◦ ◦• • ◦ ◦ ◦• • • ◦ ◦• • • • ◦} {{ }n+1Using this, give a justification of the sum of squares formula by considering the followingpicture, in which the triangles on the right are 60 ◦ -rotated versions of each other:12 23 3 34 4 4 4···n n n ··· n+n··· n4 ··· n3 4 ··· n2 3 4 ··· n1 2 3 4 ··· n+nn ···n ··· 4n ··· 4 3n ··· 4 3 2n ··· 4 3 2 11.2. The number line, field and order axioms ofR1.2.1. Depict −11/6 and √ 3 on the number line.1.2.2. Using the field axioms of R, prove the following:(1) Additive inverses are unique.(2) For all a ∈ R, 0·a = 0.(3) For all a ∈ R, (−1)·a = −a.=2n+12n+1 2n+1···2n+1 2n+1 ··· 2n+11

4 1. The real number system and preliminaries1.5. Intervals1.5.1. Let A n , n ∈ N, be a collection of sets. Then ⋂ n∈NA n denotes their intersection, that is,⋂A n = {x : ∀n ∈ N, x ∈ A n },n∈Nand we use ⋃ n∈NA n to denote the union of the sets A n , n ∈ N, that is,⋃A n = {x : ∃n ∈ N such that x ∈ A n }.n∈NProve that(1) ∅ = ⋂ (0, 1 ).nn∈N(2) {0} = ⋂ [0, 1 ].nn∈N[ 1n+2 ,1− 1(3) (0,1) = ⋃ n∈N(4) [0,1] = ⋂ n∈Nn+2(− 1 n ,1+ 1 ).n].1.5.2. Let x 0 ∈ R and δ > 0. Prove that (x 0 −δ,x 0 +δ) = {x ∈ R : |x−x 0 | < δ}.1.5.3. Prove that if x,y are real numbers, then ||x|−|y|| ≤ |x−y|.1.5.4. Show that the generalized triangle inequality: if n ∈ N and a 1 ,··· ,a n are real numbers,then |a 1 +···+a n | ≤ |a 1 |+···+|a n |. We say that the numbers a 1 ,··· ,a n have the same sign ifeither of the the following two cases is true:1 ◦ a 1 ≥ 0,··· ,a n ≥ 0.2 ◦ a 1 ≤ 0,··· ,a n ≤ 0.In other words, the numbers have the same sign if on the number lie either they all lie on theright of 0 including 0, or they all lie on the left of 0 including 0. Show that equality holds in thegeneralized triangle inequality if and only if the numbers have the same sign. Hint: Consider then = 2 case first.1.6. Functions1.6.1. Let f,g : R → R be given byx ∈ R. Compute the following:(1) f(2)+g(2).(2) f(2)−2·g(2).(3) f(2)·g(2).(4) (f(2))/(g(2)).(5) f(g(2)).f(x) = 1+x,g(x) = 1−x,

6 1. The real number system and preliminariesRemark 1.1. However, R is countable. Here is a proof of the fact that [0,1] ⊂ R is uncountable.Suppose, on the contrary, that [0,1] is countable. Let x 1 ,x 2 ,x 3 ,··· be an enumeration of [0,1].For each n ∈ N, construct a subinterval [a n ,b n ] of [0,1] that does not contain x n inductively asfollows:Initially, a 0 := 0, b 0 := 1.Suppose for k ≥ 0, a k ,b k have been chosen. Choose a k+1 ,b k+1 like this:If x k+1 ≤ a k or x k+1 ≥ b k , thenIf a k < x k+1 < b k , thena k+1 := a k + b k −a k,3b k+1 := a k +2· bk −a k.3See Figure 2a k+1 := x k+1 + b k −x k+1,3b k+1 := x k+1 +2· bk −x k+1.30 1a 0 b 0x k+1 a ka k+1 bk+1 b k x k+1a k x k+1 a k+1 b k+1 b kFigure 2. Construction of [a n,b n], n ≥ 0.Then for all n ∈ N, [a n ,b n ] ≠ ∅ and x n ∉ [a n ,b n ]. Moreover,0 < a 1 < a 2 < a 3 < ··· < a n < ··· < b n < b n−1 < ··· < b 2 < b 1 < 1.Leta := supa n and b := inf b n.n∈N n∈NThena ≤ b, andso[a,b] ≠ ∅. Also, foralln ∈ N, [a,b] ⊂ [a n ,b n ]andx n ∉ [a n ,b n ]. Soforalln ∈ N,x n ∉ [a,b]. So the points in [a,b] (⊂ [0,1]) are missing from the enumeration, a contradiction!

Chapter 2Sequences2.1. Convergence of sequences2.1.1. (Convergence/divergence)(1) Prove that the constant sequence (1) n∈N is convergent.(2) Can the limit of a convergent sequence be one of the terms of the sequence?(3) If none of the terms of a convergent sequence equal its limit, then prove that the termsof the sequence cannot consist of a finite number of distinct values.(4) Prove that the sequence ((−1) n ) n∈N is divergent.12.1.2. Prove that limn→∞ n ≠ 1.2.1.3. In each of the cases listed below, give an example of a divergent sequence (a n ) n∈N thatsatisfies the given conditions. Suppose that L = 1.(1) For every ǫ > 0, there exists an N such that for infinitely many n > N, |a n −L| < ǫ.(2) There exists an ǫ > 0 and a N ∈ N such that for all n > N, |a n −L| < ǫ.2.1.4. Let S be a nonempty subset of R that is bounded above. Show that there exists a sequence(a n ) n∈N contained in S (that is, a n ∈ S for all n ∈ N) and which is convergent with limit equal tosupS.2.1.5. Let (a n ) n∈N be a sequence such that for all n ∈ N, a n ≥ 0. Prove that if (a n ) n∈N isconvergent with limit L, then L ≥ 0. .2.2. Bounded and monotone sequences2.2.1. Let (a n ) n∈N be a sequence defined byProve that (a n ) n∈N is convergent.a 1 = 1 and a n = 2n+13n a n−1 for n ≥ 2.( )bn2.2.2. If (b n ) n∈N is a bounded sequence, then prove thatnn∈Nis convergent with limit 0.7

8 2. Sequences2.2.3. (∗) If (a n ) n∈N is a convergent sequence with limit L, then prove that the sequence (s n ) n∈N ,wheres n = a 1 +···+a n, n ∈ N,nis also convergent with limit L. Give an example of a sequence (a n ) n∈N such that (s n ) n∈N isconvergent but (a n ) n∈N is divergent.2.2.4. Given a bounded sequence (a n ) n∈N , definel k = inf{a n | n ≥ k} and u k = sup{a n | n ≥ k}, k ∈ N.Show that the sequences (l n ) n∈N , (u n ) n∈N are bounded and monotone, and conclude that theyare convergent. (Their respective limits are denoted by liminf a n and limsupa n .)n→∞2.2.5. Fill in the blanks in the following proof of the fact that every bounded decreasing sequenceof real numbers converges.Let (a n ) n∈N be a bounded decreasing sequence of real numbers. Let l ∗ be the lowerbound of {a n : n ∈ N}. The existence of l ∗ is guaranteed by the of the set of real numbers.We show that l ∗ is the of (a n ) n∈N . Taking ǫ > 0, we must show that there exists a positiveinteger N such that for all n > N. Since l ∗ +ǫ > l ∗ , l ∗ +ǫ is not of {a n : n ∈ N}.Therefore there exists N with ≤ a N < . Since (a n ) n∈N is , we have for alln ≥ N that l ∗ −ǫ < l ∗ ≤ ≤ a N < l ∗ +ǫ, and so |a n −l ∗ | < ǫ. □2.2.6. Circle the convergent sequences in the following list, and find the limit in case the sequenceconverges:(1) (cos(πn)) n∈N .n→∞(2) (1+n 2 ) n∈N .(3)(4)( ) sinn.n∈Nn(1− 3n2n+1).n∈N(5) 0.9, 0.99, 0.999, ···.2.3. Algebra of limits2.3.1. Recall the convergent sequence (a n ) n∈N from Exercise 2.2.1 on page 7 defined byWhat is its limit?a 1 = 1 and a n = 2n+13n a n−1 for n ≥ 2.Hint: If (a n ) n∈N is a convergent sequence with limit L, then (a n+1 ) n∈N is also a convergentsequence with limit L.2.3.2. Suppose that the sequence (a n ) n∈N is convergent, and assume that the sequence (b n ) n∈Nis bounded. Prove that the sequence (c n ) n∈N defined byis convergent, and find its limit.c n = a nb n +5na 2 n +n

2.4. Sandwich Theorem 92.3.3. Let (a n ) n∈N be a convergent sequence with limit L and suppose that a n ≥ 0 for all n ∈ N.Prove that the sequence ( √ a n ) n∈N is also convergent, with limit √ L.Hint: First show that L ≥ 0. Let ǫ > 0. If L = 0, then choose N ∈ N large enough so thatfor n > N, |a n − L| = a n < ǫ 2 . If L > 0, then choose N ∈ N large enough so that for n > N,| √ a n − √ L|| √ a n + √ L| = |a n −L| < ǫ √ L.2.3.4. Show that ( √ n 2 +n−n) n∈N is a convergent sequence and find its limit.Hint: ‘Rationalize the numerator’ by using √ n 2 +n+n.2.3.5. Provethatif(a n ) n∈N and(b n ) n∈N areconvergentsequencessuchthatforalln ∈ N, a n ≤ b n ,thenHint: Use Exercise 2.1.5 on page 7.lim a n ≤ lim b n.n→∞ n→∞2.4. Sandwich Theorem( n!2.4.1. Prove that the sequencen n )n∈Nn!is convergent and limn→∞ n n = 0.Hint: Observe that 0 ≤ n!n n = 1 n · 2n ····· nn ≤ 1 n ·1·····1 ≤ 1 n .( 1 k +2 k +3 k +···+n k2.4.2. Prove that for all k ∈ N, the sequence1 k +2 k +3 k +···+n klimn→∞ n k+2 = 0.n k+2 )n∈Nis convergent and2.4.3. ( limn→∞ n 1 n = 1.)(1) Using induction, prove that if x ≥ −1 and n ∈ N, then (1+x) n ≥ 1+nx.(2) Show that for all n ∈ N, 1 ≤ n 1 n < (1+ √ (n) 2 n ≤ 1+ √ 1 ) 2. nHint: Take x = 1 √ nin the inequality above.(3) Prove that (n 1 n) n∈N is convergent and find its limit.2.4.4. Let(a n ) n∈N beasequencecontainedintheinterval(a,b)(thatis, foralln ∈ N, a < a n < b).If (a n ) n∈N is convergent with limit L, then prove that L ∈ [a,b].Hint: Use Exercise 2.1.5 on page 7.Give an example to show that L needn’t belong to (a,b).2.4.5. Let (a n ) n∈N be a convergent sequence, and let (b n ) n∈N satisfy |b n −a n | < 1 nShow that (b n ) n∈N is also convergent. What is its limit?for all n ∈ N.Hint: Observe that − 1 n +a n < b n < a n + 1 for all n ∈ N.n

10 2. Sequences2.5. Subsequences, Bolzano-WeierstrassTheorem, Cauchy sequences2.5.1. Determine if the following statements are true or false.(1) Every subsequence of a convergent real sequence is convergent.(2) Every subsequence of a divergent real sequence is divergent.(3) Every subsequence of a bounded real sequence is bounded.(4) Every subsequence of an unbounded real sequence is unbounded.(5) Every subsequence of a monotone real sequence is monotone.(6) Every subsequence of a nonmonotone real sequence is nonmonotone.(7) If every subsequence of a real sequence converges, the sequence itself converges.(8) If for a real sequence (a n ) n∈N , the sequences (a 2n ) n∈N and (a 2n+1 ) n∈N both converge,then (a n ) n∈N converges.(9) If for a real sequence (a n ) n∈N , the sequences (a 2n ) n∈N and (a 2n+1 ) n∈N both converge tothe same limit, then (a n ) n∈N converges.2.5.2. (∗) Beginning with 2 and 7, the sequence 2,7,1,4,7,4,2,8,2,8,... is constructed by multiplyingsuccessive pairs of its terms and adjoining the result as the next one or two members ofthe sequence depending on whether the product is a one- or two-digit number. Thus we start with2 and 7, giving the product 14, and so the next two terms are 1,4. Proceeding in this manner, weget subsequent terms as follows:2,72,7,1,42,7,1,42,7,1,4,72,7,1,4,72,7,1,4,7,42,7,1,4,7,42,7,1,4,7,4,2,82,7,1,4,7,4,2,82,7,1,4,7,4,2,8,2,8Prove that this sequence has the constant subsequence 6,6,6,.....Hint: Show that 6 appears an infinite number of times as follows. Since the terms 2,8,2,8 areadjacent, they give rise to the adjacent terms 1,6,1,6 at some point, which in turn give rise tothe adjacent terms 6,6,6 eventually, and so on. Proceeding in this way, find out if you get a loopcontaining the term 6.2.5.3. Show that if (a n ) n∈N is a sequence that does not converge to L, then there exists an ǫ > 0and there exists a subsequence (a nk ) k∈N of (a n ) n∈N such that for all k ∈ N, |a nk −L| ≥ ǫ.2.5.4. (∗) Consider the bounded divergent sequence ((−1) n ) n∈N . Note that there exist two subsequences(−1,−1,−1,... and 1,1,1,...) which ave distinct limits (−1 ≠ 1). In this exercise weshow that this is a general phenomenon. Show that if (a n ) n∈N is bounded and divergent, then

2.6. Pointwise versus uniform convergence 11it has two subsequences which converge to distinct limits. Hint: Use the Bolzano-Weierstrasstheorem twice.2.5.5. Show that if (a n ) n∈N is a Cauchy sequence, then (a n+1 −a n ) n∈N converges to 0.2.6. Pointwise versus uniform convergence2.6.1. Suppose that I is an interval and f n : I → R (n ∈ N) be a sequence which is pointwiseconvergent to f : I → R. Let the numbers a n := sup{|f n (x) − f(x)| : x ∈ I} (n ∈ N) all exist.Prove that (f n ) n∈N converges uniformly to f if and only if limn→∞ a n = 0.Let f n : [1,∞) → R be given by f n (x) = xe −nx for x ∈ (0,∞) and n ∈ N. Show that thesequence (f n ) n∈N converges uniformly on (0,∞).2.6.2. Let f n : [0,1] → R be defined byf n (x) =Does (f n ) n∈N converge uniformly on [0,1]?x1+nx(x ∈ [0,1]).2.6.3. Let f n : R → R be defined byf n (x) = 1−1(1+x 2 ) n(x ∈ R, n ∈ N).Show that the sequence (f n ) n∈N of continuous functions converges pointwise to the function{ 1 if x ≠ 0,f(x) =0 if x = 0,which is discontinuous at 0.2.6.4. (To be done after the relevant topics have been covered.) Uniform convergence oftenimplies that the limit function inherits properties possessed by the terms of the sequence. Forinstance, we will see later on that if a sequence of continuous functions converges uniformly to afunction f, then f is also continuous. Morally, the reason nice things can happen with uniformconvergence is that we can exchange two limiting processes, which is not allowed always whenone just has pointwise convergence. The following exercises demonstrate the precariousness ofexchanging limiting processes arbitrarily.(1) For n ∈ N and m ∈ N, set a m,n = mm+n .Show that for each fixed n, lim a m,n = 1, while for each fixed m, lim a m,n = 0.m→∞ n→∞Is lim lim a m,n = lim lim a m,n?m→∞n→∞ n→∞m→∞ (2) Let f n : R → R be defined byf n (x) = sin(nx) √ n(x ∈ R, n ∈ N).Show that (f n ) n∈N converges pointwise to the zero function f. However, show that(f ′ n ) n∈N does not converge pointwise to (the zero function) f ′ .(3) Let f n : [0,1] → R (n ∈ N) be defined by f n (x) = nx(1 − x 2 ) n (x ∈ [0,1]). Show that(f n ) n∈N converges pointwise to the zero function f. However, show thatlimn→∞∫ 10f n (x)dx = 1 ∫ 12 ≠ 0 = lim f(x)dx.n→∞0

12 2. SequencesRemark 2.1. Besides the preservation of continuity under uniform convergence, one also has thefollowing results associated with uniform convergence, which we won’t establish in this course (butinstead we refer the interested student to Rudin’s book Principles of Mathematical Analysis fordetails).Proposition 2.2. If f n : [a,b] → R (n ∈ N) is a sequence of Riemann-integrable functions on[a,b] which converges uniformly to f : [a,b] → R, then f is also Riemann-integrable on [a,b], andmoreover∫ baf(x)dx = limn→∞∫ baf n (x)dx.Proposition 2.3. Let f n : (a,b) → R (n ∈ N) be a sequence of differentiable functions on (a,b),such that there exists a point c ∈ (a,b) for which (f n (c)) n∈N converges. If the sequence (f ′ n ) n∈Nconverges uniformly to g on (a,b), then (f n ) n∈N converges uniformly to a differentiable functionf on (a,b), and moreover, f ′ (x) = g(x) for all x ∈ (a,b).

Chapter 3Continuity3.1. Definition of continuity3.1.1. Given ǫ > 0, find some δ > 0 such that if x ∈ R satisfies |x−2| < δ, then1∣x − 1 2∣ < ǫ.Conclude that x ↦→ 1 : (0,∞) → R is continuous at 2.x3.1.2. Let f : R → R be a function that satisfies f(x+y) = f(x)+f(y) for all x,y ∈ R.(1) Suppose that f is continuous at some real number c. Prove that f is continuous on R.Hint: Since f is continuous at c, given ǫ > 0, ∃δ > 0 such that for all x ∈ R satisfying|x−c| < δ, |f(x)−f(c)| < ǫ. Show that given any other point c ′ ∈ R, the function f iscontinuous at c ′ by showing that the same δ works (for this ǫ).(2) Give an example of such a function.3.1.3. Suppose that f : R → R and there exists a M > 0 such that for all x ∈ R, |f(x)| ≤ M|x|.Prove that f is continuous at 0. Hint: Find f(0).3.1.4. Let f : R → R be defined byf(x) ={ 0 if x is rational,1 if x is irrational.Prove that for every c ∈ R, f is not continuous at c.Hint: Use the fact that there are irrational numbers arbitrarily close to any rational number andrational numbers arbitrarily close to any irrational number.3.1.5. Let f : R → R be a continuous function. Prove that if for some c ∈ R, f(c) > 0, then thereexists a δ > 0 such that for all x ∈ (c−δ,c+δ), f(x) > 0.3.2. Continuous functions at a point preserveconvergence of sequences there3.2.1. Let c ∈ R, δ > 0 and f : (c−δ,c] → R be continuous and strictly increasing on (c−δ,c].Show that f is strictly increasing on (c−δ,c].13

14 3. Continuity3.2.2. Prove that if f : R → R is continuous and f(x) = 0 if x is rational, then f(x) = 0 for allx ∈ R. Revisit Exercise 3.1.4.Hint: Given any real number c, there exists a sequence of rational numbers (r n ) n∈N that convergesto c.3.2.3. Let f : R → R be a function that preserves divergent sequences, that is, for every divergentsequence (x n ) n∈N , (f(x n )) n∈N is divergent as well. Prove that f is one-to-one.Hint: Let x 1 ,x 2 be distinct real numbers, and consider the sequence x 1 ,x 2 ,x 1 ,x 2 ,....3.2.4. Let I be an interval, c ∈ I, and f : I → R. Show that the following are equivalent:(1) f is continuous at c.(2) Foreverysequence(x n ) n∈N containedin I suchthat (x n ) n∈N convergesto c, the sequence(f(x n )) n∈N converges.3.2.5. Consider the function f : R → R defined by{ x if x is rational,f(x) =−x if x is irrational.Prove that f is continuous only at 0. Hint: For every rational number, there is a sequence ofirrational numbers that converges to it, and for every irrational number, there is a sequence ofrational numbers that converges to it.3.2.6. (∗) Everynonzerorationalnumber x can be uniquely written as x = n/d, where n,d denoteintegers without any common divisors and d > 0. When r = 0, we take d = 1 and n = 0. Considerthe function f : R → R defined by⎧⎪⎨ 0 if x is irrational,(f(x) = 1⎪⎩ if x = n )is rational.d dProvethat f is discontinuousat everyrationalnumber, and continuous at everyirrationalnumber.Hint: For an irrational number x, given any ǫ > 0, and any interval (N,N +1) containing x, showthat there are just finitely many rational numbers r in (N,N +1) for which f(r) ≥ ǫ. Use this toshow the continuity at irrationals.3.2.7. Let f : R → R be a continuous function such that for all x,y ∈ R, f(x+y) = f(x)+f(y).Show that there exists a real number a such that for all x ∈ R, f(x) = ax. Hint: Show first thatfor natural numbers n, f(n) = nf(1). Extend this to integers n, and then to rational numbersn/d. Finally use the density of Q in R to prove the claim.3.2.8. Find all continuous functions f : R → R such that for all x ∈ R, f(x)+f(2x) = 0.( x( x( xHint: Show that f(x) = −f = f = −f = ···.2)4)8)3.2.9. Give an example of(1) an interval I,(2) a continuous function f : I → R, and(3) a Cauchy sequence (x n ) n∈Nfor which (f(x n )) n∈N is not a Cauchy sequence in R. Compare with Exercise 3.4.4.

3.3. Intermediate and Extreme Value Theorems 153.3. Intermediate and Extreme Value Theorems3.3.1. Let f : [0,∞) → R be the function defined byf(x) = 11+x2, x ∈ R.Show that f is strictly decreasing and that f([0,∞)) = (0,1]. Find an expression for the inversefunction f −1 : (0,1] → [0,∞) and explain why f −1 is continuous on (0,1]. Sktech the graphs of fand f −1 .3.3.2. Consider a flat pancake of arbitrary shape. Show that there is a straight line cut thatdivides the pancake into two parts having equal areas. Can the direction of the straight line cutbe chosen arbitrarily?3.3.3. In each case, give an example of a continuous function f : S → R, such that f(S) = T orelse explain why there can be no such f.(1) S = (0,1), T = (0,1].(2) S = (0,1), T = (0,1)∪(1,2).(3) S = R, T = Q.(4) S = [0,1] ⋃ [2,3], T = {0,1}.3.3.4. A function f : R → R is called periodic if there exists a T > 0 such that for all x ∈ R,f(x+T) = f(x). If f : R → R is continuous and periodic, then prove that f is bounded, that is,the set S = {f(x) | x ∈ R} is bounded.3.3.5. Let f : [a,b] → R be continuous on [a,b], and define f ∗ as follows:{ f(a) if x = a,f ∗ (x) =max{f(y) : y ∈ [a,x]} if x ∈ (a,b].(1) Show that f ∗ is a well-defined function.(2) (∗) Prove that f ∗ is continuous on [a,b].(3) If f : [−1,1] → R is given by f(x) = x−x 2 , then find f ∗ .3.3.6. Supposethatf : [0,1] → Risacontinuousfunctionsuchthatforallx ∈ [0,1],0 ≤ f(x) ≤ 1.Prove that there exists at least one c ∈ [0,1] such that f(c) = c.Hint: Consider the continuous function g(x) = f(x)−x, and use the intermediate value theorem.

16 3. Continuity3.3.7. At 8:00 a.m. on Saturday, a hiker begins walking up the side of a mountain to his weekendcampsite. On Sunday morning at 8:00 a.m., he walks back down the mountain along the sametrail. It takes him one hour to walk up, but only half an hour to walk down. At some point on hisway down, he realizes that he was at the same spot at exactly the same time on Saturday. Provethat he is right.Hint: Let u(t) and d(t) be the position functions for the walks up and down, and apply theintermediate value theorem to f(t) = u(t)−d(t).3.3.8. Show that the polynomial function p(x) = 2x 3 −5x 2 −10x+5has a real root in the interval[−1,2].3.3.9. Let f : R → R be continuous. If S := {f(x) : x ∈ R}is neither bounded abovenor boundedbelow, prove that S = R.Hint: If y ∈ R, then since S is neither bounded above nor bounded below, there exist x 0 ,x 1 ∈ Rsuch that f(x 0 ) < y < f(x 1 ).3.3.10. (∗) Show that given any continuous function f : R → R, there exists a x 0 ∈ [0,1] and am ∈ Z\{0} such that f(x 0 ) = mx 0 . In other words, the graph of f intersects some nonhorizontalline y = mx at some point x 0 in [0,1].Hint: If f(0) = 0, take x 0 = 0 and any m ∈ Z\{0}. If f(0) > 0, then choose N ∈ N satisfyingN > f(1), and apply the intermediate value theorem to the continuous function g(x) = f(x)−Nxon the interval [0,1]. If f(0) < 0, then first choose a N ∈ N such that N > −f(1), and considerthe function g(x) = f(x)+Nx, and proceed in a similar manner.3.3.11. (∗) Prove that there does not exist a continuous function f : R → R such that assumesrational values at irrational numbers, and irrational values at rational numbers, that is,f(Q) ⊂ R\Q and f(R\Q) ⊂ Q.Hint: Note that for every m ∈ Z\{0}, there does not exist a x 0 ∈ R such that f(x 0 ) = mx 0 .3.4. Uniform continuity3.4.1. Show that f : R → R given by f(x) = x 2 (x ∈ R) is continuous, but not uniformlycontinuous. Hint: Consider x = n and y = n+ 1 nfor large n.3.4.2. Prove that the function f : R → R defined by f(x) = |x| (x ∈ R) is uniformly continuous.3.4.3. A function f : R → R is said to be (globally) Lipschitz if there exists a number L > 0 suchthat for all x,y ∈ R, |f(x) −f(y)| ≤ L|x−y|. Prove that every Lipschitz function is uniformlycontinuous.3.4.4. Let I be an interval and let f : I → R be uniformly continuous. Show that if (x n ) n∈N is aCauchy sequence, then (f(x n )) n∈N is a Cauchy sequence. Compare this with Exercise 3.2.9.3.5. Limits3.5.1. Let f(a,c)∪(c,b) → R be such that lim f(x) and lim f(x) exist. Show that lim f(x)x→c+ x→c− x→cexists if and only if lim f(x) = lim f(x).x→c+ x→c−

3.5. Limits 173.5.2. In each of the following cases, calculate the limit if it exists.|x|(1) limx→0 x+1 .(2) limx→1(⌊x⌋−x).(3) limx→0x⌊x⌋.(4) limx→0sin 1 x .3.5.3. Let the polynomials A,B of degrees α,β ≥ 1 be given bywhere a α and b β are nonzero. Show thatA(x)limx→∞A(x) = a 0 +a 1 x+···+a α x α ,B(x) = b 0 +b 1 x+···+b β x β ,0 if α > β,a α⎧⎪ ⎨ if α = β,B(x) = b β⎪ ⎩+∞ if α < β and a αb β> 0,−∞ if α < β and a αb β< 0.

Chapter 4Differentiation4.1. Definition of the derivative4.1.1. Use the definition of the derivative to find f ′ (x), where f(x) := √ x 2 +1, x ∈ R.4.1.2. If f : (a,b) → R is differentiable at c ∈ (a,b), then show thatexists and equals f ′ (c). Is the converse true?f(c+h)−f(c−h)limhց0 2h4.1.3. Let f : (0,∞) → R be a function and let c > 0. Show that the following are equivalent:(1) f is differentiable at c.f(kc)−f(c)(2) lim exists.k→1 k −1Moreover, show that if either (1) or (2) hold, then f ′ (c) = 1 c limk→1f(kc)−f(c).k −14.1.4. Let f : (0,∞) → R be such that for all x,y ∈ (0,∞), f(xy) = f(x) + f(y). If f isdifferentiable at 1, then show that f is differentiable at every c ∈ (0,∞) and that f ′ (c) = f ′ (1)/c.Conclude that f is infinitely differentiable. If f ′ (1) = 2, then find f (n) (3), n ∈ N.4.1.5. (Differentiable but not continuously differentiable example.) Consider f : R → R definedby f(0) = 0 and for x ≠ 0,f(x) = x 2 sin 1 x .Prove that f is differentiable, but f ′ is not continuous at 0.4.2. The Product, Quotient and Chain Rule4.2.1. Prove that if f,g are infinitely differentiable on an open interval I, thenn∑( n(fg) (n) (x) = fk)(k) (x)g (n−k) (x), x ∈ I.k=0For a rational x and n a nonnegative integer, define x [n] := x(x−1)···(x−n+1). Show that ifx,y ∈ Q, thenn∑( n(x+y) [n] = xk)[k] y [n−k] .k=019

20 4. DifferentiationHint: Differentiate t x+y n times with respect to t ∈ I := (0,∞).(In fact after we have defined logarithms, and how to take real exponents, that is, the mapt ↦→ t x : (0,∞) → R, where x ∈ R, one can see that the same result holds even when x,y ∈ R.)4.2.2. Let f : (0,∞) → R be the strictly decreasing function given byf(x) = 11+x2, x ∈ (0,∞).From Exercise 3.3.1, it follows that f((0,∞)) = (0,1). Show that the inverse f −1 : (0,1) → (0,∞)is differentiable, and find the derivative (f −1 ) ′ .4.2.3. Find f ′ (x) for each of the following functions:(1) f(x) = sin(xsinx)+sin(sin(x 2 )).(2) f(x) = sin(6cos(6sin(6cosx))).( ((3) f(x) = sin (sin(sinx)) 2)) 3.(You may use the fact that sin ′ = cos and cos ′ = −sin.)4.2.4. By using the binomial expansion (1+x) n =(1)(2)(3)(4)n∑( ) n3 k .kn∑( ) nk 2 .kn∑)1 n.k +1(kn∑( n(2k +1) .k)k=1k=1k=1k=14.2.5. Evaluate the sumk=1n∑k=0( nk)x k , find expressions forn∑ k 2n 2 k by considering the function S given by S(x) = ∑x k .k=14.3. Tangents and normals to curves,Newton-Raphson Method4.3.1. Find the values of the constants a,b,c for which the graphs of f,g given byf(x) := x 2 +ax+b,g(x) := x 3 −c,x ∈ R, intersect at the point (1,2) and have the same tangent there.

4.4. Mean Value Theorem, Rolle’s Theorem, monotonicity, Taylor’s Formula 21( 1)4.3.2. Find the equation of the tangent at4 ,4 to the curve t ↦→ (x(t),y(t)), wherefor t ∈ (0,∞).x(t) := 1 t 2,y(t) := √ t 2 +12,4.3.3. Find the points on the curve given implicitly by x 2 +xy +y 2 = 7 at which(1) the tangent is parallel to the x-axis(2) the tangent is parallel to the y-axis.4.3.4. Find the tangents to the implicitly defined curve x 2 y + xy 2 = 6 at the points for whichx = 1. Also compute d2 yat these points.dx2 4.3.5. Considerthe function f givenbyf(x) = x 2 −2. Suppose thatthe Newton-RaphsonMethodconverges with the initial guess x 0 := 1. Generate a few rational approximations to √ 2.4.3.6. Show that x 4 −x 3 −75 = 0 for some x between 3 and 4. Use the Newton-Raphson Methodto find an approximate value of this x, starting with an initial guess of x 0 := (3+4)/2 = 3.5.4.4. Mean Value Theorem, Rolle’s Theorem,monotonicity, Taylor’s Formula4.4.1. Show that for every real a,b ∈ R, |cosa−cosb| ≤ |a−b|.4.4.2. Suppose that f : R → R is differentiable, |f ′ (x)| ≤ 1 for all x ∈ R, and that there exists ana > 0 such that f(−a) = −a, f(a) = a. Find f(0). Give an example of such a function f.4.4.3. Suppose that f : R → R is differentiable and that there are L,L ′ ∈ R such thatProve that L ′ = 0.lim f(x) = L,n→∞limn→∞ f′ (x) = L ′ .4.4.4. Let c ∈ (a,b), and let f : (a,b) → R be such that f is(1) differentiable on (a,b)\{c},(2) continuous on (a,b), and(3) limx→cf ′ (x) exists.Then f is differentiable at c, and f ′ (c) = limx→cf ′ (x).4.4.5. Let f : R → R be such that for all x,y ∈ R, |f(x) − f(y)| ≤ (x − y) 2 . Prove that f isconstant.4.4.6. Let f : (a,b) → R be differentiable on (a,b) and suppose that there is number M such thatfor all x ∈ (a,b), |f ′ (x)| ≤ M. Show that f is uniformly continuous on (a,b).

22 4. Differentiation4.4.7. Find all functions f : R → R such that f is differentiable on R and for all x ∈ R and alln ∈ N,f ′ (x) = f(x+n)−f(x) .nHint: Conclude that f must be twice differentiable and calculate f ′′ (x).4.4.8. Prove that if c 0 ,··· ,c d are any real numbers satisfyingc 01 + c 12 +···+ c dd+1 = 0,then the polynomial c 0 +c 1 x+···+c d x d has a zero in (0,1).4.4.9. Show that if f : (a,b) → R has a local minimum at c ∈ (a,b), and if f is twice differentiableat c, then f ′′ (c) ≥ 0.Suppose that F satisfies the differential equationF ′′ (x)+F ′ (x)g(x)−F(x) = 0for some function g. Prove that if F is 0 at two points, then F is 0 on the interval between them.4.4.10. Suppose that f is n times differentiable and that f(x) = 0 for n + 1 distinct x. Provethat f (n) (x) = 0 for some x.4.4.11. Show that there are exactly two real values of x such thatand that they lie in4.4.12. (On y ′′ +y = 0.)(− π 2 , π ).2x 2 = xsinx+cosx(1) If y = y(x) is a solution of the differential equationthen show that y 2 +(y ′ ) 2 is constant.y ′′ +y = 0, (4.1)(2) Use part (1) to show that every solution to (4.1) has the formy(x) = Acosx+Bsinxfor suitable constants A,B. Proceed as follows: It is easy to show that all functions ofthe aboveform satisfy (4.1). Let y be a solution. For it to have the form Acosx+Bsinx,it is necessary that A = y(0) and B = y ′ (0). Now considerf(x) := y(x)−y(0)cosx−y ′ (0)sinx,and apply (1) to f, making use of the fact that f(0) = f ′ (0) = 0.(3) Use part (2) to prove the trigonometric addition formulaesin(a+b) = (sina)(cosb)+(cosa)(sinb),cos(a+b) = (cosa)(cosb)−(sina)(sinb).4.4.13. Find the shortest distance from a given point (0,b) on the y-axis with b > 0, to theparabola y = x 2 .4.4.14. Does f given by f(x) = (sinx−cosx) 2 , x ∈ R have a maximum value? If so, find it.4.4.15. Use Taylor’s Formula to show that limx→0sinx−xx 3 = − 1 6 .

4.5. Optimisation, convexity 234.4.16. (o-notation) A special notation introduced by Landau in 1909 is particularly suited toTaylor’s formula. Given functions f,g in an open interval I containing a such that g is nonzero inI, we write“f(x) = o(g(x)) as x → a”f(x)if limx→a g(x) = 0.The symbol f(x) = o(g(x)) is read “f(x) is little-oh of g(x)” or “f(x) is of smaller order thang(x)”, and is intended to convey the idea that for x near a, f(x) is small as compared to g(x).For example,f(x) = o(1) as x → a means that lim f(x) = 0,x→aandf(x)f(x) = o(x) as x → a means that limx→a x = 0.Also, if h is a function on I, then we write “f(x) = g(x) + o(h(x)) as x → a” to mean that“f(x)−g(x) = o(h(x)) as x → a”.(1) If f has a continuous (n+1)st derivative in some open interval containing the compactinterval [a−ǫ,a+ǫ], and ifM := max{|f n+1 (x)| : |x−a| ≤ ǫ},then it follows from Taylor’s Formula thatn∑ f (k) (a)f(x) = (x−a) k +o(x n ) as x → a.k!k=0(2) Show that tanx = x+ x33 +o(x3 ) as x → 0.tanx−x(3) Show that limx→0 sinx−xcosx = 1.4.4.17. Let f : R → R. We call x ∈ R a fixed point of f if f(x) = x.(1) If f is differentiable, and for all x ∈ R, f ′ (x) ≠ 1, then prove that f has at most onefixed point.(2) (∗) Show that if there is an M < 1 such that for all x ∈ R, |f ′ (x)| ≤ M, then there isa fixed point x ∗ of f, and that x ∗ = lim x n, where x 1 is arbitrary and x n+1 = f(x n )n→∞(n ∈ N).(3) Visualize the process described in the part (2) above via the zig-zag path(x 1 ,x 2 ) −→ (x 2 ,x 2 ) −→ (x 2 ,x 3 ) −→ (x 3 ,x 3 ) −→ (x 3 ,x 4 ) −→ ....(4) Prove that the function f : R → R defined byf(x) = x+ 11+e x (x ∈ R)has no fixed point, although 0 < f ′ (x) < 1 for all x ∈ R. Is this a contradiction to theresult in part (2) above? Explain.4.5. Optimisation, convexity4.5.1. Sketchthecurvey = 2x 3 +2x 2 −2x−1afterlocatingintervalsofincrease/decrease,intervalsof convexity/concavity 1 , points of maxima/minima, and points of inflection (places where f ′′ = 0).How many times, and approximately where does the curve cross the x-axis?1 A function is concave if −f is convex.

24 4. Differentiation4.5.2. If a 1 < ··· < a n , then find the minimum value ofn∑f(x) := (x−a k ) 2 , x ∈ R.Next find the minimum value ofg(x) :=k=1n∑|x−a k |, x ∈ R.k=1This is a problem where Calculus won’t help: on the intervals between the a k , the function islinear, and so the minimum must occur at one of the a k , and these are points where g is notdifferentiable. However, the answer is easy to find if one considers how f changes as one passesfrom one such interval to another.Let a > 0. Show that the maximum value of h is 2+a , where h is given by1+ah(x) =11+|x| + 11+|x−a| , x ∈ R.Hint: Consider (−∞,0), (0,a), (a,∞) separately, and find the derivative there.4.5.3. The picture in Figure 1 below shows the graph of the derivative f ′ of f. Find all localmaximizers and local minimizers of f.f ′0 123 4Figure 1. Graph of f ′ .4.5.4. A cannon ball is shot from the ground with velocity v in the vertical plane at an angleα from the horizontal, so that it has an initial vertical component of velocity equal to vsinα,and a horizontal component vcosα. Its horizontal velocity remains constant at vcosα, while thevertical component changes under gravitational force. The height h(t) at time t above the groundis given by h(t) = −5t 2 +v(sinα)t for t ≥ 0. Find the angle α that maximizes the span (that isthe horizontal distance reached) of the cannon ball.4.5.5. Show that x ↦→ |x| : R → R is convex.4.5.6. Let f : I → R be a function on an interval I ⊂ R. We define the epigraph of f byU(f) := ⋃ x∈I{(x,y) : y ≥ f(x)} ⊂ I ×R.In other words, U(f) is the “region above the graph of f”. A subset C ⊂ R 2 is called a convex setif for all v 1 ,v 2 ∈ C and for all t ∈ (0,1), (1−t)·v 1 +t·v 2 ∈ C. Show that f is a convex functionif and only if U(f) is a convex set.

4.6. 0 form of l’Hôpital’s Rule 2504.5.7. (The Cauchy-Schwarz Inequality.)(1) Let a > 0, b,c ∈ R and consider the function f : R → R given byf(t) = at 2 +bt+c, t ∈ R.Show that f has a minimizer and that the minimum value of f is − b2 −4ac.4a(2) Show that for all n ∈ N and all a 1 ,··· ,a n ,b 1 ,··· ,b n ∈ R, there holds that(a 2 1 +···+a2 n )(b2 1 +···+b2 n ) ≥ (a 1b 1 +···+a n b n ) 2 .n∑Hint: Consider (ta k −b k ) 2 .k=14.5.8. (The Arithmetic Mean-Geometric Mean Inequality.)4.6.(1) Suppose that f : I → R is a convex function on an interval I ⊂ R. If n ∈ N, andx 1 ,··· ,x n ∈ I, then show that( )x1 +···+x nf ≤ f(x 1)+···+f(x n ).n n(2) Show that −log : (0,∞) → R is convex.(3) Prove the Arithmetic Mean-Geometric Mean Inequality: for nonnegative real numbersa 1 ,··· ,a n , there holds thata 1 +···+a n≥nn√ a 1···a n .(The left hand side above is called the arithmetic mean of a 1 ,··· ,a n , while the righthand side is called their geometric mean.)00form of l’Hôpital’s Rule4.6.1. What is wrong with the following application of l’Hôpital’s Rule?x 3 +x−2limx→1 x 2 −3x+2 = limx→1Show that the limit above is actually equal to −4.3x 2 +12x−3 = limx→16x2 = 3.4.6.2. Revisit Exercise 4.4.15, but now use l’Hôpital’s Rule to show that limx→0sinx−xx 3 = − 1 6 .

Chapter 5Integration5.1. Definition and properties of the Riemannintegral5.1.1. Is the following function f Riemann integrable on [0,1]?{ 0 if x ∈ [0,1]\Q,f(x) =x if x ∈ [0,1]∩Q.Hint: For any partition P = {x 0 = 0 < x 1 < ··· < x n−1 < x n = 1}, x k+1 ≥ x k+1 +x k,2k = 0,··· ,n−1. Use this to find a positive lower bound on upper sums.5.1.2. Let f,g ∈ R[a,b]. Show that max{f,g} and min{f,g} also belong to R[a,b], wheremax{f,g} := max{f(x),g(x)},min{f,g} := min{f(x),g(x)},for all x ∈ [a,b]. Hint: max{a,b} = a+b+|a−b|2for a,b ∈ R.5.1.3. Give examples of bounded functions f,g : [0,1] → R that are not Riemann integrable, but|f|, f +g, fg are all Riemann integrable.5.1.4. (An integral mean value result.) Let f ∈ C[a,b], ϕ ∈ R[a,b], and let ϕ be pointwisenonnegative. Use the Intermediate Value Theorem for f to show that there is a c ∈ [a,b] suchthat∫ baf(x)ϕ(x)dx = f(c)∫ baϕ(x)dx.Give examples to show that neither the continuity of f nor the nonnegativity of ϕ can be omitted.5.2. Fundamental Theorem of Integral Calculus5.2.1. Evaluate limx→0+∫1 xx 3 0t 2t 4 +1 dt.x5.2.2. If f is continuous on R, then evaluate limx→x 0 x 2 −x 2 0and x 0 = 0 separately.∫ xx 0f(t)dt. Consider the cases x 0 ≠ 027

28 5. Integration5.2.3. Let T > 0, and f : R → R be a continuous function which is T-periodic, that is, f(x+T) =f(x) for all x ∈ R. Show that the integralhas the same value for all a ∈ R.∫ a+Taf(x)dx5.2.4. (Leibniz’s Rule for Integrals.) If f ∈ C[a,b] and u,v are differentiable on [c,d] andu([c,d]) ⊂ [a,b], v([c,d]) ⊂ [a,b], thenddx∫ v(x)u(x)5.2.5. For x ∈ R, definef(t)dt = f(v(x))·v ′ (x)−f(u(x)·u ′ (x), x ∈ [c,d].Find F ′ and G ′ .F(x) :=G(x) :=∫ 2x1∫ x2011+t 2dt,11+ √ |t| .5.2.6. Let f : [0,∞) → R be continuous. Find f(2) if for all x ≥ 0,(1)(2)(3)(4)∫ x0∫ f(x)0∫ x2f(t)dt = x 2 (1+x).0∫ x 2 (1+x)0t 2 dt = x 2 (1+x).f(t)dt = x 2 (1+x).f(t)dt = x.5.2.7. Let f be a continuous function on R and λ ≠ 0. Considery(x) = 1 λ∫ x0f(t)sin(λ(x−t))dt for x ∈ R.Show that y is a solution to the inhomogeneous differential equation y ′′ (x)+λ 2 y(x) = f(x) for allx ∈ R and with the initial conditions y(0) = 0 and y ′ (0) = 0.5.3. Integration by Parts and by Substitution5.3.1. Let a > 0 and f ∈ C[−a,a] be an odd function, that is, f(x) = −f(−x) for all x ∈ [−a,a].Prove that∫ a−af(x)dx = 0.5.3.2. Let m,n be nonnegative integers. Show that x m (1−x) n m!n!dx =0 (m+n+1)! .Hint: If I(m,n) denotes the integral, then show that I(m,n) = n I(m+1,n−1) for n ∈ N.m+1∫ 1

5.5. Improper integrals 295.3.3. For f ∈ C[a,b], show that for all x ∈ [a,b],∫ xa(∫ ua)f(t)dt du =∫ x0(x−u)f(u)du.Remark 5.1. More generally, one can show, using induction, that∫ x(∫ un( (∫ u1) ) ∫ xf(u)·(x−u)··· f(t)dt)du n1 ··· du n = du.aa n!aa5.3.4. (Taylor’s Formula with Integral Remainder.) Let n be a nonnegative integer and f ∈C n+1 [a,b]. Show thatf(b) = f(a)+f ′ (a)(b−a)+···+ f(n) (a)(b−a) n + 1 n! n!∫ ba(b−t) n f (n+1) (t)dt.(Note that as opposed to the Taylor’s Formula we have met before, where the error term containedan undetermined c, now the “integral remainder” does not involve such an undetermined numberc.)∫ 1nx n−15.3.5. Find limn→∞0 1+x dx.5.3.6. Evaluate∫ 140x√1−4x2 dx.5.3.7. Evaluate∫ 813√ 3√x·√x4 −1dx.5.4. Riemann sums(15.4.1. Find lim √n→∞ 12 +n + 1√ 2 22 +n + 1√ 2 32 +n +···+ 1√).2 n2 +n 25.5. Improper integrals5.5.1. Determine whether or not the following improper integrals exist:(1)(2)∫ ∞0∫ ∞01√1+x3 dx.x1+ √ x 3dx.5.5.2. (Properties of the Gamma function Γ.)(1) Show that Γ(1) = 1.(2) For s > 0, Γ(s+1) = s·Γ(s).(3) Show that for all n ∈ N, Γ(n+1) = n!.

30 5. Integration5.5.3. Suppose that f : [0,∞) → [0,∞) is such that∫ ∞0f(x)dx (5.1)exists. Intuitively, we expect that the areaunder the graph of the nonnegative f in intervals [x,∞)to become smaller and smaller as x becomes larger and larger, and so one is tempted to concludethat f itself must have limit 0 as x → ∞. Show with an example that this needn’t be the case.Suppose now that we know that f : [0,∞) → [0,∞) is such that, besides having that theimproper integral (5.1) exists, also f is differentiable and thatexists. Show that lim f(x) = 0.x→∞∫ ∞0f ′ (x)dx5.5.4. (Convolution) For f,g : R → R, which are both zero outside some compact interval, wedefine the convolution f ∗g : R → R by(f ∗g)(t) =∫ ∞assuming that the integral exists for each t.−∞f(τ)g(t−τ)dτ, t ∈ R,(1) Note that the graph of g(−·) is obtained by reflecting the graph of g about the y-axis,and for a fixed t, the graph of g(t −·) is a shifted version of the graph of g(−·). So inorder to find out the value (f ∗g)(t), one may proceed as follows.(a) Draw the graph of f and g.(b) Reflect the graph of g about the y-axis.(c) Translate the graph of g(−·) by |t| units to the left if t < 0, and to the right if t ≥ 0.(d) Multiply the functions f and g(t−·) pointwise, and find the area under the graphof this pointwise product.Use this procedure to graphically determine the convolution 1 [0,1] ∗1 [0,1] , where 1 [0,1] isthe indicator function of the interval [0,1]:{ 1 if x ∈ [0,1],1 [0,1] (x) =0 if x ∈ R\[0,1].(2) Show that f ∗g = g ∗f.5.5.5. (Differentiation under the integral sign. 1 ) While we don’t learn the theory behind this, letus try to use this tool formally, that is, without paying attention to rigour. To findwe consider the more general integralI(α) =∫ ∞0∫ ∞0sinxx dx,e −αxsinxx dx,1 Differentiation under the integral sign is mentioned in Feynman’s memoir Surely You’re Joking, Mr. Feynman!in the chapter “A Different Box of Tools”, where he mentions learning it from a Calculus book by Woods while in highschool. “But I caught on how to use that method, and I used that one damn tool again and again. So because I wasself-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princetonhad trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned inschool. If it was contour integration, they would have found it; if it was a simple series expansion, they would have foundit. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation fordoing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on itbefore giving the problem to me.”

5.5. Improper integrals 31by introducing the “parameter” α. (So our integral of interest corresponds to α = 0.) Then bydifferentiating with respect to α we obtain∫ ∞I ′ (α) = (−x)e −αxsinx∫ ∞0 x dx = −e −αx sinxdx = − 10 α 2 +1 e−αx ((−α)sinx−cosx) ∣ ∞ 0= − 1α 2 +1 .Thus0−I(α) = I(∞)−I(α) = − π 2 +tan−1 α,and sointegrals.(1)(2)(3)(4)∫ ∞0∫ ∞0∫ ∞0∫ 10∫ ∞0sinxx dx = I(0) = π . Try your hand at formally finding the values of the following2(sinx) 2x 2 dx by considering I(α) =∫ ∞e −xsinx∫ ∞x dx by considering I(α) =∫x−1∞logx dx by considering I(α) =000(sin(αx)) 2dx.x 2e −xsin(αx) dx.xx α −1logx dx.tan −1 (πx)−tan −1 xdx by considering I(α) =x5.5.6. (Gravitational potential energy and escape velocity.)∫ ∞0tan −1 (αx)−tan −1 xdx.x(1) The gravitationalpotential energy(due to the gravitationalpull ofthe Earth)at distancea R from the center of the earth can be thought to be the amount of work done to bringan object from separation R to far away (“at infinity R = ∞”). By Newton’s Law ofGravitation, the force experienced by a mass m at a separation r from the center of theEarth is given byF = GMmr 2 ,where G is the universal gravitation constant and M is the mass of the Earth. The workdone to move a mass m at r through a small distance dr is given byWork done = (Force)·(Displacement) = GMm dr,where we have assumed that the force is constant over [r,r + dr] for small dr. So thepotential energy V(R) at R is given by the (improper) integralV(R) := − =Find an explicit expression for V(R).∫ ∞RGMmr 2 dr.(2) The escape velocity v e (R) at a separation R from the center of the earth is defined as tobe the one which imparts enough kinetic energy to the object in order to overcome thegravitation potential energy V(R). Show that√2GMv e =R .Using the following values, determine the escape velocity of a rocket on the surface ofthe earth:Radius of the Earth = 6,371 km,Mass of the Earth = 5.97219×10 24 kg,Universal gravitational constant G = 6.67384×10 −11 m 3 kg −1 s −2 .r 2

32 5. Integration5.6. Transcendental functions5.6.1. (Euler’s constant γ.) Another important number named after Euler, is the Euler’s constantγ, which plays, among other things, a role in Number Theory.(1) Prove that 1 n ≥ log(n+1)−logn ≥ 1 for all n ∈ N.n+1(2) Ifa n := 1+ 1 2 + 1 3 +···+ 1 n −logn,then show that (a n ) n∈N is decreasing and that a n ≥ 0 for all n ∈ N. Conclude that thereis a numberγ := lim(1+ 1n→∞ 2 + 1 3 +···+ 1 )n −logn .(Approximately, γ = 0.5772156649···, but it is not known whether γ is rational orirrational. The reason behind the choice of the symbol is that γ is closely related to theΓ function. For example, γ = −Γ ′ (1).)5.6.2. Show that for all x ≥ 0, x− x22≤ log(1+x) ≤ x−x22 + x33 .5.6.3. In “hyperbolic geometry” of the unit disc D := {(x,y) ∈ R 2 : x 2 +y 2 < 1} in the plane,straight lines in D are circular arcs that are orthogonal to the bounding circle T. See Figure 1.The distance between two points A,B is then taken asTDAB CPQFigure 1. A straight line passing through A,B,C in the picture on the left. In Escher’s “CircleLimit I”, the backbones of the fish lie along hyperbolic lines, and all of the white fish are“hyperbolically congruent” to each other, as are all of the black fish.d(A,B) := log( ⌢AP⌢AQ·where ⌢ AP denotes the circular Euclidean arc length, etc. Show that for three points A,B,C lyingon such a line, d(A,B)+d(B,C) = d(A,C).5.6.4. Show that(1)(2)∫ ba∫ ba⌢BQ⌢BPlogxdx = blogb−aloga+a−b for all a,b ∈ (0,∞).e x dx = e b −e a for all a,b ∈ R.),

5.6. Transcendental functions 335.6.5. (Stirling’s Formula.)(1) Compute the improper integral∫ 10logxdx.(2) Based on the result in the previous part, try giving a formal nonrigorous explanation ofStirling’s Formula: for large n, logn! ≈ nlogn−n.5.6.6. If a,b > 0 and b ≠ 1, then we define “the logarithm of a to the base b”, by log b a := logalogb .(1) Show that b log b a = a.(2) Prove that log 2 3 is irrational.(3) Show that there are irrational numbers a,b such that a b is rational.(4) Find the flaw in the following argument given for the claim that 1 > 2.“We know that 4 > 2. As the logarithm is a strictly increasing function, takinglog 1/2 of both sides, we obtain −2 = log 1/2 4 > log 1/2 2 = −1, and so 1 > 2.”(5) Sketch the graphs of x ↦→ logx, log 1/2 x, log 10 x in the same picture.5.6.7. Show that for all x ∈ R,∫ x0∫ x01√1+t2 dt = log(x+√ 1+x 2 ),√1+t2 dt = x√ 1+x 2 +log(x+ √ 1+x 2 ).25.6.8. Let a,x i ∈ R, and consider the “Initial Value Problem”{ x ′ (t) = ax ′ (t), t ≥ 0,x(0) = x i ,Show that this Initial Value Problem has a unique solution, given by x(t) = e ta x i , t ≥ 0.Hint: To show uniqueness, consider the derivative of f, where f(t) := e −ta x(t).5.6.9. (Newton’s Law of Cooling.) Newton’s Law of Cooling states that an object cools at a rateproportional to the difference of its temperature and the temperature of the surrounding medium.Find the temperature Θ(t) of an object at time t, in terms of the its temperature Θ 0 at time0, assuming that the temperature of the surrounding medium is kept at a constant, M. Whathappens as t → ∞?Hint: To solve the differential equation, note that (e kt (Θ − M)) ′ = e kt (Θ ′ + k(Θ − M)) for aconstant k.5.6.10. (Radioactive decay.) A radioactive substance diminishes at a rate proportional to theamount present (since all atoms have equal probability of disintegrating, the total disintegrationis proportionalto the number ofatomsremaining). IfA(t) is the amountat time t, this means thatA ′ (t) = −cA(t) for some c > 0 (which represents the probability that an atom will disintegrate).(1) Find A(t) in terms of the amount A(0) = A 0 present at time 0.(2) Show that there is a number τ (the half-life of the radioactive element) with the propertythat A(t+τ) = A(t)/2 for all t.

34 5. Integration5.6.11. (Compound interest.) An amount of P SEK is deposited in a bank which pays an interestat a rate rper year, compounded m times a year. Thus the total amount (of principal plus interest)at the end of n years is(A = P 1+m) r mn.If r,n are kept fixed, show that this amount approaches the limit Pe rn as m → ∞. This motivatesthe following definition. We say that the money grows at an annual rate r when compoundedcontinuously if the amount A(t) after t years is Pe rt for all t ≥ 0. Approximately how long doesit take for a bank account to double in value if it receives interest at an annual rate of 6% if(1) compounded continuously?(2) it is just a simple interest?5.6.12. (The hyperbolic functions.) The hyperbolic sin and hyperbolic cos functions sinh,cosh :R → R are defined asfor x ∈ R.coshx := ex +e −x,2sinhx := ex −e −x,2(1) Show that for any t ∈ R, the point (cosht,sinht) is on the hyperbola x 2 −y 2 = 1. Sketchthe portion of the hyperbola obtained.(2) Show also thatsinh0 = 0, cosh0 = 1,sinh ′ x = coshx, cosh ′ x = sinhx, x ∈ R,cosh(x+y) = (coshx)(coshy)+(sinhx)(sinhy), x,y ∈ R.(3) Sketch the graphs of sinh and cosh.5.6.13. Show that limx→0+ xx = 1. Conclude that limn→∞ n1/n = 1.5.6.14. Show that the improper integralNote that for x > 0,∫ 101x x = e−xlogx =1xxdx exists.∞∑ (−xlogx) n.n!Based on this, try giving a formal argument to justify the curious identity∫ 10n=01∞x xdx = ∑ 1n n.n=1((5.6.15. Find lim 1+ 1 )(1+ 2 ) (··· 1+ nn→∞ n n n) ) 1 n.5.6.16. Order the following functions from the fastest growing to the slowest growing:2 x , e x , x x , (logx) x , e x/2 , x 1/2 , ,log 2 x, log(logx), (logx) 2 , x e , x 2 ,logx, (2x) x , x 2x .

5.6. Transcendental functions 355.6.17. Evaluate the following limits:(1) limx→∞log(logx).logx3 sinx −1(2) lim .x→0 x(3) limx→0sinx−x+x 3 /6x 3 .cosx−1+x 2 /2(4) limx→0 x 4 .( 1(5) limx→0 x − 1 ).sinx5.6.18. Evaluate∫ 12− 1 2(cosx)·log( ) 1−xdx.1+x5.6.19. Find the derivative of f : (0,∞) → R, wheref(x) = logxx , x > 0.Which is bigger: e π or π e ? (You may use the estimates e < 3 < π.)5.6.20. Prove that if m,n ∈ N, then∫ π−π∫ π−π(sin(mx))(sin(nx))dx =(cos(mx))(sin(nx))dx = 0.{ 0 if m ≠ n,π if m = n}=∫ π−π(cos(mx))(cos(nx))dx,5.6.21. (Fourier Series.) Let n ∈ N and a 0 ,a 1 ,··· ,a n and b 1 ,··· ,b n be real numbers. Considerthe function f : R → R defined byn∑f(x) := a 0 + (a k cos(kx)+b k sin(kx)), x ∈ R.k=1(1) Show that f is 2π-periodic, that is, f(x+2π) = f(x) for all x ∈ R.(2) Prove thatand for k = 1,··· ,n,a kb ka 0 = 1 ∫ πf(x)dx,2π −π= 1 π= 1 π∫ π−π∫ π−πf(x)cos(kx)dx,f(x)sin(kx)dx.(3) Let a k := 0 for 0 ≤ k ≤ n, and⎧⎪⎨4if k is even,b k := kπ⎪⎩ 0 otherwise.Take n = 3, n = 33 and n = 333, successively, and in each case, plot the resulting fusing a package such as Maple, Mathematica or Matlab on the computer. What do youobserve?

36 5. Integration5.6.22. (Fixed points of sin and cos.)(1) Show that 0 is the only fixed point of sin : R → R.(2) Prove that cos : R → R has a unique fixed point c ∗ .(3) Determine experimentally the value of c ∗ as follows. In your scientific calculator, key inany number, and press the “cos key” repeatedly. After a while, the display stabilizes.Explain by means of a picture, why this displayed value must be c ∗ (approximately).5.6.23. (Addition formula for tan.)(1) Show that for real x,y such that x,y,x+y are not in πZ+ π , there holds that2tan(x+y) = tanx+tany1−(tanx)(tany) .(2) Prove that tan1 ◦ ∉ Q. (Here one degree, 1 ◦ , is the the angle measuring π180 radians.)5.6.24. Show thatUsing the fact that e ={ 1 if x ∈ Q,lim limm→∞ n→∞ (cos(2πm!x))n =0 if x ∉ Q.∞∑n=01n!and the above result, show that e ∉ Q.5.6.25. (Visual differentiation of tan.) We have learnt thattan ′ x =1(cosx) 2 = 1+(tanx)2 . (5.2)Here we offer a formal geometric explanation. Consider the triangle shown in Figure 2. If weldxdtanxx1dxltanxFigure 2. Visual differentiation of tan.increase x by a small amount dx, then tanx increases by the amount dtanx shown in the figure.Show that (5.2) holds by imagining that in the limiting case when dx diminishes to 0, the bluetriangle is ultimately similar to the red one.5.6.26. Find∫ 1011+t 2dt.

5.7. Applications: length, area, volume 375.6.27. Show that∫ π2011+(tanx) √ 2 dx = π 4 .(The integrand is taken as 1 when x = 0, and 0 when x = π 2 .)1( πHint: Consider the symmetry in the graph of1+(tanx) √ about 2 4 , 1 ).25.6.28. For x ∈ (−1,1), the inverse sin −1 : (−1,1) → (−π/2,π/2) of the strictly increasingcontinuous function sin : (−π/2,π/2) → (−1,1) exists. Prove that(sin −1 ) ′ (y) =1√1−y2 , y ∈ (−1,1).For y ∈ (−1,1), show that∫ y0∫ y01√1−t2 dt = sin−1 y,√1−t2 dt = y√ 1−y 2 +sin −1 y.25.6.29. Find the polar coordinates of the following points given in Cartesian/rectangular coordinates:(1) (1,1).(2) (1,0).(3) (0,1).(4) (−1,0).(5) (−1,−1).(6) (0,−1).5.7. Applications: length, area, volume5.7.1. One winter morning in Lund, it started snowing at a heavy and constant rate. A snowploughstarted out at 8 : 00 a.m. At 9 : 00 a.m. it had gone 2 swedish miles. By 10 : 00 a.m., ithad gone 3 swedish miles. Assuming that the snowplough removes a constant volume of snow perhour, determine the time at which it started snowing.5.7.2. Find the area enclosed by an ellipse described bya 2 + y2b 2 = 1,where a,b > 0. What happens when a = b?x 25.7.3. The horizontal line y = c intersects the curve y = 2x−3x 3 in the first quadrant as shownin the figure below. Find c so that the areas of the two shaded regions are equal.5.7.4. Calculate the areaenclosed by the two “petals”ofthe lemniscate givenin polar coordinatesby r 2 = 2cos(2θ).5.7.5. Calculate the volume of a doughnut, with the radius of the greater circle equal to R (thatis, of the central circle lying midway in the annular region obtained by taking a horizontal crosssection of the doughnut), and that of the two little circles, obtained by taking a vertical crosssection of the doughnut, equal to r.

38 5. Integration2x 3 −3x 2y = cFigure 3. The two shaded areas.5.7.6. Calculate the volume of an ellipsoid, namely the solid of revolution obtained by revolvingthe region enclosed by the ellipsea 2 + y2b 2 = 1,where a,b > 0, in the upper half-plane and the x-axis. What happens when a = b?x 25.7.7. A round hole of radius √ 3 is bored through the center of a solid ball of radius 2 cm. Findthe volume cut out.5.7.8. The position of a particle in the plane R 2 at time t ≥ 0 is given byx(t) = 1 3 (2t+3)1/3 ,y(t) = t2 2 +t,for t ≥ 0. Find the distance it travels between t = 0 and t = 3. What is its average speed?5.7.9. Calculate the arclength of the cardioid given in polar coordinates by r = 2(1+cosθ).5.7.10. (The Elliptic Integral.) Show that the perimeter of an ellipse given bywhere b ≥ a > 0, is given byb∫ 2π0x 2a 2 + y2b 2 = 1,√1−k2 (sinθ) 2 dθ,for a suitable constant k. (This integral is called an elliptic integral, and cannot be expressed interms of elementary functions when b ≠ a.) What happens when b = a?5.7.11. Calculate the surface area of a doughnut, with the radius of the greater circle equal to R(that is, of the central circle lying midway in the annular region obtained by taking a horizontalcross section of the doughnut), and that of the two little circles, obtained by taking a vertical crosssection of the doughnut, equal to r.5.7.12. How accurately should we measure the radius of a ball in order to calculate its surfacearea within 3% of its exact value?

5.7. Applications: length, area, volume 395.7.13. Let a ∈ R with a > 0. An arc of the “catenary” given by( xy = acosa)whose end points have x-coordinates 0 and a is revolved about the x-axis. Show that the surfacearea A and the volume V of the solid thus generated are related by the formulaA = 2V a .5.7.14. (Design of a clepsydra.) A clepsydra (literally meaning “water thief” in Greek) or a waterclock is designed by revolving the graph of x ↦→ Cx m : [0,r] → R, where C,m > 0, as shown inFigure 4.Cx mrFigure 4. Clepsydra.Whatshouldmbeifthelevelofwateristodecreaselinearlyastimepasses? YoumayuseToricelli’sLaw stating that the speed of water flowing out when the height of water is h is proportional to√h.

Chapter 6Series6.1. Convergence/divergence of series6.1.1. (Tantalising tan −1 .) Show thatHint: Write∞∑n=1tan −1 12n 2 = π 4 .12n 2 = (2n+1)−(2n−1)tana−tanband use tan(a−b) =1+(2n+1)(2n−1) 1+tanatanb .6.1.2. Show that for every real number x > 1, the seriesconverges. Hint: Add11+x + 21+x 2 + 4 2n+···+ +...1+x4 1+x 2n11−x .6.1.3. Consider the Fibonnaci sequence (F n ) n∈N with F 0 = F 1 = 1 and F n+1 = F n + F n−1 forn ∈ N. Show that∞∑ 1= 1.F n−1 F n+16.1.4. Does the series∞∑n=1n=2( 1cos converge?n)6.1.5. Prove that if a 1 ≥ a 2 ≥ a 3 ... is a sequence of nonnegative numbers, and if∞∑a n < +∞,n=1then a n approaches 0 faster than 1/n, that is, limn→∞ na n = 0.Hint: Consider the inequalities a n+1 +···+a 2n ≥ n·a 2n and a n+1 +···+a 2n+1 ≥ n·a 2n+1 .Show that the assumption a 1 ≥ a 2 ≥ a 3 ... above cannot be dropped by considering thelacunary series whose n 2 th term is 1/n 2 and all other terms are zero.6.1.6. Consider the Arithmetic-Geometric progression1, 2r, 3r 2 , 4r 3 , ···41

42 6. Serieswhere r ∈ R. Note that 1,2,3,4,··· form an arithmetic progression, while 1,r,r 2 ,r 3 ,··· form ageometric progression. Show that if |r| < 1, then1+2r+3r 2 +4r 3 +··· =6.2. Absolute convergence and the Leibniz’sAlternating Series Test6.2.1. Does the series∞∑n=1sinnn 2 converge?1(1−r) 2.∞∑ (−1) n6.2.2. Let s > 0. Show thatn=1n sconverges.6.2.3. Prove that6.2.4. Prove that∞∑√ n(−1) n n+1 converges.n=1∞∑(−1) n sin 1 n converges.n=16.3. Comparison, Ratio, Root6.3.1. Determine if the following series are convergent or not.∞∑ n 2(1)2 n.(2)(3)n=1∞∑ (n!) 2(2n)! .∞∑( ) n 4n 5 .5n=1n=16.3.2. Prove that∞∑n=1nn 4 +n 2 +1Hint: Write the denominator as (n 2 +1) 2 −n 2 .6.3.3. Let (a n ) n∈N be a real sequence such thatHint: First conclude that for large n, |a n | < 1.6.3.4. (∗) We have seen that the seriesconverges. (∗) Can you also find its value?∞∑n=1∞∑∞ a 4 n converges. Show that ∑a 5 n converges.convergesif and only if p > 1. Suppose that we sum the series for all n that can be written withoutusing the numeral 9. (Imagine that the key for 9 on the keyboard is broken.)n=11n pCall the resulting summation ∑ ′. Prove that the series∑ ′ 1n pn=1

6.3. Comparison, Ratio, Root 43converges if and only if p > log 10 9. Hint: First show that there are 8·9 k−1 numbers without 9between 10 k−1 and 10 k .6.3.5. Determine if the following statements are true or false.∞∑∞∑(1) If |a n | is convergent, then so is a 2 n .(2) Ifn=1∞∑a n is convergent, then so isn=1(3) If limn→∞ a n = 0, thenn=1∞∑a 2 n .n=1∞∑a n converges.n=1(4) If limn→∞ (a 1 +···+a n ) = 0, then(5)∞∑n=1log n+1nconverges.∞∑a n converges.n=1(6) If a n > 0 (n ∈ N) and the partial sums of (a n ) n∈N are bounded above, thenconverges.(7) If a n > 0 (n ∈ N) and∞∑a n converges, thenn=1∞∑n=11a ndiverges.∞∑n=1a n6.3.6. (Fourier series.) In order to understand a complicated situation, it is natural to try tobreak it up into simpler things. For example, from Calculus we learn that an analytic functioncan be expanded into a Taylor series, where we break it down into the simplest possible analyticfunctions, namely monomials 1,x,x 2 ,... as follows:f(x) = f(0)+f ′ (0)x+ f′′ (0)x 2 +....2!The idea behind the Fourier series is similar. In order to understand a complicated periodicfunction, we break it down into the simplest periodic functions, namely sines and cosines. Thusif T ≥ 0 and f : R → R is T-periodic, that is, f(x) = f(x+T) (x ∈ R), then one tries to findcoefficients a 0 ,a 1 ,a 2 ,... and b 1 ,b 2 ,b 3 ,... such thatf(x) = a 0 +∞∑n=1(a n cos( 2πnT x )+b n sin( 2πnT x )). (6.1)Suppose that the Fourier series given in (6.1) converges pointwise to f on R. Show that if∞∑(|a n |+|b n |) < ∞,n=1then in fact the series converges uniformly.The aim of this part of the exercise is to give experimental evidence for two things. Firstly,the plausibility of the Fourier expansion, and secondly, that the uniform convergence might fail ifthe condition in the previous part of this exercise does not hold. To this end, let us consider thesquare wave f : R → R given by{ 1 if x ∈ [n,n+1) for n even,f(x) =−1 if x ∈ [n,n+1) for n odd.

44 6. SeriesThen f is a 2-periodic signal. From the theory of Fourier Series, which we will not discuss in thiscourse, the coefficients can be calculated, and they happen to be 0 = a 0 = a 1 = a 2 = a 3 = ... and⎧⎪⎨4if n is odd,b n = nπ⎪⎩ 0 if n is even.Write a Maple program to plot the graphs of the partial sums of the series in (6.1) with, say, 3,33, 333 terms. Discuss your observations.Figure 1. Partial sums of the Fourier series for the square wave.6.3.7. Show that if∞∑a n converges, where each a n is nonnegative, then so doesn=16.3.8. Let a n ≥ 0 for all n ∈ N. Prove that∞∑a n converges if and only ifn=1∞∑n=1∞∑ √an a n+1 .n=1a n1+a nconverges.6.3.9. Let l 1 ,l 2 be defined byl 1 :=l 2 :={ }∞∑(a n ) n∈N : |a n | < ∞ ,n=1{}∞∑(a n ) n∈N : |a n | 2 < ∞ .n=1Show that l 1 ⊂ l 2 . Is l 1 = l 2 ?

6.4. Power series 456.3.10. As the harmonic series∞∑n=11n diverges, the reciprocal 1/s n of the nth partial sums n := 1+ 1 2 + 1 3 +···+ 1 napproaches 0 as n → ∞. So the necessary condition for the convergence of the series∞∑ 1sn=1 nis satisfied. But we don’t know yet whether or not it actually converges. It is clear that the harmonicseries diverges very slowly, which means that 1/s n decreases very slowly, and this promptsthe guess that this series diverges. Show that in fact our guess is correct. Hint: s n < n.6.3.11. Show that the series∞∑n=11n n converges.6.3.12. Consider the Fibonnaci sequence (F n ) n∈N with F 0 = F 1 = 1 and F n+1 = F n +F n−1 forn ∈ N. Show that∞∑ 1< +∞.F nn=0Hint: F n+1 = F n +F n−1 > F n−1 +F n−1 = 2F n−1 . Using this, show that both 1 F 0+ 1 F 2+ 1 F 4+...and 1 F 1+ 1 F 3+ 1 F 5+... converge.6.3.13. Show that∞∑k=16.4. Power series(sin π √ )k 4 +1 converges absolutely.6.4.1. Show that for all x ∈ R,sinx =∞∑(−1) n−1 x 2n−1(2n−1)!n=1cosx =∞∑(−1) n x2n(2n)!n=0x3= x−3! + x5−+···, and5!= 1−x22! + x44! −+··· .6.4.2. (A C ∞ function which is not analytic.) Let f : R → R be given by{ef(x) =−1/x2 if x ≠ 0,0 if x = 0.(1) Sketch the graph of f.f(x)(2) Prove that for every n ∈ N, limx→0 x n = 0.(3) Show that for each n ∈ N, there exists a polynomial p n such that for all nonzero x,( 1f (n) (x) = e −1/x2 p .x)(4) Provethat f (n) (0) = 0foralln ≥ 1(showingthatthe Taylorseriesforf at0isidenticallyzero, clearly not equal to f).