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(1) In this problem you will work with a new definition. Given a sequence {an}0, we define its buddy sequence to be the sequence {An}-o defined Šn=0 as follows: Vn E N, An — ап+1 — аn. For the duration of this problem only, we always use lowercase letters for sequences and uppercase letters for their buddy sequences, as defined above. You may do the same in your solutions without having to define the buddy sequence every time. has all negative (a) Let {bn}, be a sequence with all positive terms. Assume that {Bn}o terms. Prove that {bn}_o converges. (b) Prove that if a sequence is convergent, then its buddy sequence converges to zero. (c) Let {an}, be a sequence. Assume that {A„}o converges to zero. Is it necessarily true that {an}o converges? If so, prove it. If not, give a counterexample and justify why it works. In=0 n=D0

(1) In this problem you will work with a new definition. Given a sequence {an}0, we define its buddy sequence to be the sequence {An}-o defined Šn=0 as follows: Vn E N, An — ап+1 — аn. For the duration of this problem only, we always use lowercase letters for sequences and uppercase letters for their buddy sequences, as defined above. You may do the same in your solutions without having to define the buddy sequence every time. has all negative (a) Let {bn}, be a sequence with all positive terms. Assume that {Bn}o terms. Prove that {bn}_o converges. (b) Prove that if a sequence is convergent, then its buddy sequence converges to zero. (c) Let {an}, be a sequence. Assume that {A„}o converges to zero. Is it necessarily true that {an}o converges? If so, prove it. If not, give a counterexample and justify why it works. In=0 n=D0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(1) In this problem you will work with a new definition. Given a sequence {an}0, we define its buddy sequence to be the sequence {An} as follows: defined Vn E N, An = an+1 – An. For the duration of this problem only, we always use lowercase letters for sequences and uppercase letters for their buddy sequences, as defined above. You may do the same in your solutions without having to define the buddy sequence every time. (a) Let {bn}, be a sequence with all positive terms. Assume that {Bn}, has all negative terms. Prove that {bn}o converges. n=0 (b) Prove that if a sequence is convergent, then its buddy sequence converges to zero. (c) Let {an}, be a sequence. Assume that {An}o converges to zero. Is it necessarily true that {an}o converges? If so, prove it. If not, give a counterexample and justify why it works. 100 n=0 100
Transcribed Image Text:(1) In this problem you will work with a new definition. Given a sequence {an}0, we define its buddy sequence to be the sequence {An} as follows: defined Vn E N, An = an+1 – An. For the duration of this problem only, we always use lowercase letters for sequences and uppercase letters for their buddy sequences, as defined above. You may do the same in your solutions without having to define the buddy sequence every time. (a) Let {bn}, be a sequence with all positive terms. Assume that {Bn}, has all negative terms. Prove that {bn}o converges. n=0 (b) Prove that if a sequence is convergent, then its buddy sequence converges to zero. (c) Let {an}, be a sequence. Assume that {An}o converges to zero. Is it necessarily true that {an}o converges? If so, prove it. If not, give a counterexample and justify why it works. 100 n=0 100
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