Homework 11
Math 501
Due November 21, 2014
Exercise 1
Let (an ) ⊂ R be a sequence such that an ≥ 0 for all n and
Prove that
∞ √
X
an
n
n=1
P∞
n=1
an converges.
converges. [Hint: First show that if x, y > 0 then xy ≤ 12 (x2 + y 2 ).]
Exercise 2
Prove that the series
∞
X
1
p
n
log
n
n=1
converges when p > 1 and diverges when p ≤ 1. [Note: logp n simply means
(log n)p in the same way sin2 x means (sin x)2 . Moreover, log means the natural
logarithm with base e.]
Exercise 3
Prove the positive part of the limit comparison test. That is, suppose that
(an ), (bn ) ⊂ R are sequences of positive real numbers and
an
< ∞.
bn
P∞
P∞
Prove that if n=1 bn converges, then n=1 an converges.
lim sup
n→∞
Exercise 4
P∞
Let (an ) ⊂ R be a sequence
of a positive real numbers. Prove that n=1 an
P∞
converges if and only if n=1 log(1 + an ) converges. [Hint: use L’Hôpital’s rule
and the limit comparison test.]
1
Exercise 5
Q∞
An infinite product is a formal product n=1 cn whose factors are positive
real numbers. The nth partial product is Cn = c1 · · · cn . If Cn converges to a
limit C 6= 0, then the infinite product is said to converge to C.
Q∞Suppose cn = 1 + an where anP≥∞0 for all n or an ≤ 0 for all n. Prove that
n=1 cn converges if and only if
n=1 an converges.
Exercise 6
P∞
Let (an ), (bn ) ⊂ R be sequences of real numbers.
P∞ Prove that if n=1 |an |
converges and (bn ) is a bounded sequence, then n=1 an bn converges.
Exercise 7
Let fn , f : [a, b] → R be functions. Suppose that fn → f pointwise and
Z
b
Z
fn dx →
a
f dx
a
as n → ∞. Prove or disprove: fn ⇒ f .
2
b