Math 231 Midterm Exam 2 (Solutions)
Prof. I.Kapovich October 19, 2009
Problem 1.[20 points]
For each of the following statements indicate whether it is true or false.
You DO NOT need to provide explanations for your answers in this problem.
(1) If limn→∞ an= 0 then the series P∞
n=1 anconverges.
(2) If a series P∞
n=1 andiverges then the series P∞
n=1 |an|also diverges.
(3) If P∞
n=1(an+bn) converges then the series P∞
n=1 anand P∞
n=1 bn
both converge.
(4) Whenever anis a monotone increasing sequence of real numbers then
there exists A∈Rsuch that limn→∞ an=A.
(5) If an≥0 for all n≥1 and the series P∞
n=1 anconverges, then the
series P∞
n=1 ansin(an) also converges.
Answers:
(1) FALSE. For example, limn→∞
1
n= 0 but the series P∞
n=1 1
ndiverges.
(2) TRUE. If P∞
n=1 |an|converges, then, by Theorem 3 in Ch 11.6, the
series P∞
n=1 analso converges, contrary to our assumption that P∞
n=1 an
diverges.
(3) FALSE. For example, P∞
n=1(n+ (−n)) = P∞
n=1 0 converges but the
series P∞
n=1 nand P∞
n=1(−n) both diverge.
(4) FALSE. For example, the sequence an=nis positive and monotone
increasing, but limn→∞ n=∞ 6∈ R.
(5) TRUE. Indeed, |ansin(an)| ≤ |an|=an. Therefore by the Comparison
Test the series P∞
n=1 |ansin(an)|converges and hence, by Theorem 3 in Ch
11.6, the series P∞
n=1 ansin(an) also converges.
Problem 2.[20 points]
For each of the following series determine if it converges conditionally,
converges absolutely or diverges. Show all the details of your work.
(a)
∞
X
n=2
(−1)n
nln n
(b)
∞
X
n=1
(−1)n(n+ 1)n
(2n+ 1)n
Solution.
(a) Since the sequence bn=1
nln nsatisfies bn≥0, limn→∞
1
nln n= 0 and
is bnmonotone decreasing for n≥2, the series P∞
n=2
(−1)n
nln nconverges by
the Alternating Series Test. We have |(−1)n
nln n|=1
nln n≥0 for n≥2. To
1
