1. Decide whether the following statements are True or False. Provide a short justification for each conclusion. (a) If every subsequence of (a,) converges, then (a,) converges as well. (b) If (a,) contains a divergent subsequence, then (a,) diverges. (c) If r is areal number, then (d) Every Cauchy sequence is convergent. 2. Using the definition of a Cauchy sequence, prove that (1/n2) is a Cauchy sequence. 3. Calculate the following series if they exist. Otherwise, explain why a series diverges. Mention any relevant theorems or ideas you use. (e) È(-)" 1 (a) E k(k +3) (4) E) (b) k=1 4. Determine whether the following series converge of diverge. If the series converges, then determine the sum. Mention any relevant theorems or ideas you use. (a) k=1 (b) E(-1)"()" e (c) E(-1)" 1 (d) E k(k +3) WI

1. Decide whether the following statements are True or False. Provide a short justification for each conclusion. (a) If every subsequence of (a,) converges, then (a,) converges as well. (b) If (a,) contains a divergent subsequence, then (a,) diverges. (c) If r is areal number, then (d) Every Cauchy sequence is convergent. 2. Using the definition of a Cauchy sequence, prove that (1/n2) is a Cauchy sequence. 3. Calculate the following series if they exist. Otherwise, explain why a series diverges. Mention any relevant theorems or ideas you use. (e) È(-)" 1 (a) E k(k +3) (4) E) (b) k=1 4. Determine whether the following series converge of diverge. If the series converges, then determine the sum. Mention any relevant theorems or ideas you use. (a) k=1 (b) E(-1)"()" e (c) E(-1)" 1 (d) E k(k +3) WI

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Description: 1. Decide whether the following statements are True or False. Provide a short justificationfor each conclusion.(a) If every subsequence of (a,) converges, then (a,) converges as well.(b) If (a,) contains a divergent subsequence, then (a,) diverges.(

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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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

1. Decide whether the following statements are True or False. Provide a short justification
for each conclusion.
(a) If every subsequence of (a,) converges, then (a,) converges as well.
(b) If (a,) contains a divergent subsequence, then (a,) diverges.
(c) If r is areal number, then
(d) Every Cauchy sequence is convergent.
2. Using the definition of a Cauchy sequence, prove that (1/n2) is a Cauchy sequence.
3. Calculate the following series if they exist. Otherwise, explain why a series diverges.
Mention any relevant theorems or ideas you use.
(e) È(-)"
1
(a) E
k(k +3)
(4) E)
(b)
k=1
4. Determine whether the following series converge of diverge. If the series converges, then
determine the sum. Mention any relevant theorems or ideas you use.
(a)
k=1
(b) E(-1)"()"
e
(c) E(-1)"
1
(d) E
k(k +3)
WI

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