Final Exam - Kalamazoo College
Final Exam - Kalamazoo College
Final Exam - Kalamazoo College
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Math 320, Real Analysis I<br />
<strong>Final</strong> <strong>Exam</strong><br />
Part III: Taylor Series and Lagrange’s Remainder Theorem<br />
These questions are worth 10 points each, for a total of 30 points.<br />
Let f be infinitely differentiable on the interval (−R, R). Define a n = f (n) (0)<br />
n!<br />
let<br />
S N (x) = a 0 + a 1 x + a 2 x 2 + · · · + a N x N .<br />
for each n, and<br />
The polynomial S N (x) is a partial sum of the Taylor series expansion for the function f(x). Define<br />
E N (x) = f(x) − S N (x)<br />
which represents the error between f and the partial sum S N . Clearly proving S N (x) → f(x) is<br />
equivalent to showing E N (x) → 0.<br />
1. Explain why the error function E N (x) = f(x) − S N (x) satisfies<br />
for all n = 0, 1, 2, . . . , N.<br />
E (n)<br />
N (0) = 0<br />
2. Assume x > 0 and consider the functions E N (x) and x N+1 on the interval [0, x]. Show that<br />
there exists a point x 1 ∈ (0, x) such that<br />
E N (x)<br />
x N+1 = E′ N (x 1)<br />
(N + 1)x N 1<br />
3. Prove Lagrange’s Remainder Theorem, which states:<br />
Let f be infinitely differentiable on (−R, R), define a n = f (n) (0)/n!, and let<br />
S N (x) = a 0 + a 1 x + a 2 x 2 + · · · + a N x N .<br />
Given x ≠ 0, there exists a point c satisfying |c| < |x| where the error function<br />
E N (x) = f(x) − S N (x) satisfies<br />
.<br />
E N (x) = f (N+1) (c)<br />
(N + 1)!<br />
x N+1 .<br />
Part IV: Arzelà-Ascoli Theorem<br />
Question 1 is worth 5 points while the remaining questions are 10 points each, so this section is<br />
worth a total of 45 points.<br />
Recall the Bolzano-Weierstrass Theorem (Theorem 2.5.5) states that every bounded sequence<br />
of real numbers has a convergent subsequence. An analogous statement for bounded sequences<br />
of functions is not true in general. (See Part I, Problem 4.) Under stronger hypotheses,<br />
however, we can derive a Bolzano-Weierstrass Theorem for sequences of functions. First, we make<br />
the following definition:<br />
A sequence of functions (f n ) defined on a set E ⊆ R is called equicontinuous if for every<br />
ε > 0 there exists a δ > 0 such that<br />
for all n ∈ N and |x − y| < δ in E.<br />
|f n (x) − f n (y)| < ε