MAT 137Y: Calculus with proofs
Assignment 9
Due on Thursday, March 25 by 11:59pm via Crowdmark
Instructions:
• You will need to submit your solutions electronically via Crowdmark. See MAT137
Crowdmark help page for instructions. Make sure you understand how to submit
and that you try the system ahead of time. If you leave it for the last minute and
you run into technical problems, you will be late. There are no extensions for any
reason.
• You may submit individually or as a team of two students. See the link above for
more details.
• You will need to submit your answer to each question separately.
• This problem set is about Unit 13.
1. (a) Prove the following Theorem
Theorem 1. Let
∞∑
n=1
an be a series.
IF
 limk→∞
2k∑
n=1
an exists
lim
n→∞
an = 0
THEN the series
∞∑
n=1
an is convergent.
Hint: You may use results from past assignments. They will make things simpler.
(b) Prove that the theorem is false if we remove any one of the two hypotheses.
2. Let
∞∑
n
an be a CONVERGENT, NON-NEGATIVE series. Let f is a continuous function
with domain R. Decide whether each of the following series must be convergent, must be
divergent, or we do not have enough information to decide. Prove it.
(a)
∞∑
n
(nn · an)
(b)
∞∑
n
ln (2 + an)
(c)
∞∑
n
ln
2 + an+1
2 + an
(d)
∞∑
n
ln (1 + an)
(e)
∞∑
n
(−1)n√an
(f)
∞∑
n
(an f(sinn)) 