Math 3100 - Practice Test II.
Problem 1. For each of the following series state whether it converges
absolutely, converges conditionally or diverges. Justify your answers!
(a) P∞
n=1
cos(nπ)
n
(b) P∞
n=1
cos(n)
n2
(c) P∞
n=1 sin(1/n2)
(d) P∞
n=1
ln (n)
n3/2
Hint: Prove, and use the fact that: limn→∞
ln (n)
np= 0 for every p > 0)
(e) P∞
n=1 10n
√n!
(f) P∞
n=1
n100
√2n
Problem 2. Prove the following statements.
(a) If anis bounded and P∞
n=1 bnis absolute convergent, then P∞
n=1 anbn
is also absolute convergent.
(b) If {an>0},{bn>0}, and limn→∞
an
bn= 1, then
∞
X
n=1
anconvergent ≡∞
X
n=1
bnconvergent
(c) If {an>0},{bn>0},{an}is bounded and P∞
n=1 bnis convergent,
then P∞
n=1 anbnis convergent.
(d) If P∞
n=1 anconverges, then P∞
n=1 a2
nconverges, as well.
Problem 3. Provide counterexamples to the following false statements.
(a) If limn→∞ an= 0 then P∞
n=1 anconverges.
(b) If limn→∞ |an+1
an|= 1, then P∞
n=1 andiverges.
(c) If P∞
n=1 anconverges, then P∞
n=1 |an|converges.
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