It is important to distinguish between ∑n=1∞nb and ∑n=1∞bn What name is given to the first series? To the second? For what values of b does the first series converge? For what values of b does the second series converge?

Question

It is important to distinguish between  ∑n=1∞nb and ∑n=1∞bn  What name is given to the first series? To the second? For what values of b does the first series converge? For what values of b does the second series converge?

SkladanH · Accepted Answer

Given: The series ∑n=1∞nb and ∑n=1∞bn As, ∑n=1∞nb=1b+2b+3b+4b+... Thus it is an arithmetic increasing power series. Similarly, ∑n=1∞bn=b+b2+b3+b4+... Here first term is b and common ratio is also b. Thus it is a geometric power series. Ratio test: Consider the series: ∑n=1∞an Let L=limn→∞|an+1an| If L&amp;lt; 1, then series is convergent. If L&amp;gt;1, then series is divergent and If L=1, then test is inconclusive. For ∑n=1∞nb the nth term an=nb Thus L=limn→∞|(n+1)bnb| =limn→∞|(n+1n)b| =limn→∞|(1+1n)b| =(1+0)b =1 As L=1, test is inconclusive. Series is convergent if L&amp;lt;1. But for any value of b it will never be less than 1. Therefore, for any value of b, the series ∑n=1∞nb is not convergent. For ∑n=1∞bn the nth term of series is an=bn Thus L=limn→∞|bn+1bn| =limn→∞|b| =b Series is convergent if L&amp;lt;1. Thus b&amp;lt;1. Therefore, for b&amp;lt;1, the series ∑n=1∞bn is convergent.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

It is important to distinguish between sum_{n=1}^{infty} n^{b} quad text { and } quad sum_{n=1}^{infty} b^{n} What name is given to the first series? To the second? For what values of b does the first series converge? For what values of b does the second series converge?

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Answer & Explanation

New Questions in Calculus 2