Final Exam - Kalamazoo College

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