MAT 137 - ASSIGNMENT # 9
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MAT 137 - ASSIGNMENT # 9
DUE MARCH 19
Instructions
Please submit solutions to the questions listed below on Crowdmark. Your assignment is due before 11.59pm on March 19. You should have received an email from Crowdmark containing the assignment questions and a link you can use to submit your solutions. You will need to submit a separate file for each question, so it is easiest to write up your solution to each question on its own paper. Please see the “Crowdmark How-To” document on the course page for tips about simplifying the submission process.
Your solutions should be neat and legible (preferably in black or blue ink). The written assign- ment is worth 10 points, and we will grade five (5) questions.
Technical issues will not be accepted as a reason for failing to submit on time, so please leave yourself ample time to submit your HW before the deadline.
Assigned Questions
(1) Use the definition of convergence to show that lim =
(2) Use the definition of limit of a sequence to prove the “Squeeze Theorem for Sequences”:
“Let ean}, ebn}, ecn} be sequences. If an < bn < cn for all n > N0 and lim an = L = lim cn ,
then bn converges, and lim bn = L.”
(3) Prove that the sequence ean} given by an = converges. What is its limit?
(4) Let ean} be a sequence. A subsequence of ean} is a sequence of the form: an1 , an2 , an3 , . . . , ank , . . . where nk are natural numbers so that n1 < n2 < n3 < . . .. We will denote a subsequence by eank }.
For example: If we take an = n + 1 for n > 0, this gives the sequence 0, 2 , 3 , 4 , 5 , . . .. The sequence 0, , . . . obtained by taking every 4th term is a subsequence. So is the se- quence , , , . . . (obtained by eliminating the first two terms.)
(a) Consider the sequence with an = 1 + (-1)n ╱ 、. Find two subsequences with differ-
ent limits.
n 1 2 3 4
(b) Construct a non-constant sequence, which has a constant subsequence, or prove that such
a sequence cannot exist.
(c) Let ean} be a sequence and eank} a subsequence. Prove that if ean} converges, then eank} converges and has the same limit.
(d) Is the converse of (b) true?
(5) Let ean}, ebn} be sequences so that an+2 = 2bn for all n. Prove that ean} converges if and only if ebn} converges.
(6) Let ean} be the sequence defined recursively by a0 = 2, a1 = 1, an =
3an-1 |
an-2 + 6 . |
(a) Prove that this sequence is decreasing.
(b) Prove that this sequence is bounded (both above and below).
(c) Prove that this sequence is convergent, and find its limit.
o
(7) Use partial sums to prove that the series converges, and find its sum
o (-1)k+1 32k -1
(8) Does the series π 3k+2 converge or diverge? If it converges, find its sum.
(9) Determine if each statement is true or false. If true, give a proof. If false, explain why, or provide a counter-example.
(a) If ean} is a sequence which diverges, then the sequence ecan} diverges for all c ∈ R.
o
(b) If ean} is an increasing sequence, then an diverges.
n=0
o
(c) If ean} is a non-negative sequence then the partial sums of the series an form a
n=0
monotonic sequence.
2023-05-08