Math 501
Introduction to Real Analysis
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Summer 2015
Exam #3
Sample
This is a take-home examination. The exam includes 5 questions and one bonus problem.
The total mark is 100 points, not including the bonus question. Please show all the work,
not only the answers.
1 [20 Points]. Determine the radius of the convergence R of the following series and discuss
whether or not they converge at x = R and x = −R :
P∞ xn
√
(a)
n=1 n
(b)
P∞
(c)
P∞
(d)
P∞
n=1
(−1)n−1 xn
n4
nn
n
n=1 (n!)2 x
√
n=1
n
n2 + 2n − n xn
P
nn
2n
0
2 [20 Points]. Let f (x) = ∞
n=0 (n!)2 x , where we use the usual convention that 0 = 1.
Note that f (x) is well-defined for every x ∈ R, according to the result in Problem 1(c) above.
(a) Compute f 0 (0) or show that the derivative doesn’t exist.
(b) Prove that f is a convex function.
(c) Show that for any x > 0,
1
f (x) − 1 <
2ex
x
Z
f (x) dx <
0
1
f (x) − 1 .
x
Hint: Integrate part by part. To estimate the result of the integration you may use
without proof the following bounds:
1 n
< e,
∀ n ∈ N.
(1)
1< 1+
n
3 [20 Points].
(a) Prove that
lim
n→∞
1 Z
n!
∞
n −x
x e dx = 1.
1
1
(b) Prove or disprove: If a sequence of real-valued functions fn converges to f on [a, b],
and
Z b
Z b
f (x)dx,
fn (x)dx =
lim
n→∞
a
a
then fn converges to f uniformly on [a, b].
4 [20 points].
(a) Prove or disprove: Every point-wise converging sequence of functions fn : R → R
contains a uniformly converging subsequence.
(b) Construct sequences (fn )n∈N and (gn )n∈N of real-valued functions on R which converge
uniformly but such that the product fn gn doesn’t converge uniformly.
5 [20 points]. Let (fn )n∈N be a sequence of real-valued continuous functions which converges
uniformly on a set E. Prove that
lim fn (xn ) = f (x)
n→∞
for every sequence of points (xn )n∈N such that limn→∞ xn = x in E.
6 [Bonus question]. Prove the upper bound in (1) in the statement of Problem 2.
2