Use the comparison test or the limit comparison test to decide if each of these converges or11Σ1+ Vk3k + 5k=1k=1

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Description: Use the comparison test or the limit comparison test to decide if each of these converges or11Σ1+ Vk3k + 5k=1k=1

Answered: Use the comparison test or the limit…

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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Use the comparison test or the limit comparison test to decide if each of these converges or
1
1
Σ
1+ Vk
3k + 5
k=1
k=1

Expert Solution

Part a.)

We have,

1 < sqrt(k) or 1 + sqrt(k) < 2sqrt(k) for all k.

So, 1 / (1 + sqrt(k)) > 1 / 2sqrt(k).

Now, apply Summation from k = 1 to ♾ on both sides of the above inequality to obtain that the required series is divergent using the comparison test as series on the right side is divergent.

Note that Summation k = 1 to ♾ 1 / 2sqrt(k) is divergent as Summation k = 1 to ♾ 1/ k^pis divergent if p <_ 1, here p =1/2 < 1.

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