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?

Jaya Legge 2021-01-15 Answered
It is important to distinguish between
n=1nb and n=1bn
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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Expert Answer

SkladanH
Answered 2021-01-16 Author has 80 answers

Given:
The series n=1nb and n=1bn
As,
n=1nb=1b+2b+3b+4b+...
Thus it is an arithmetic increasing power series.
Similarly,
n=1bn=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=1an
Let L=limn|an+1an|
If L< 1, then series is convergent.
If L>1, then series is divergent and
If L=1, then test is inconclusive.
For n=1nb 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<1.
But for any value of b it will never be less than 1.
Therefore, for any value of b, the series n=1nb is not convergent.
For n=1bn the nth term of series is an=bn
Thus
L=limn|bn+1bn|
=limn|b|
=b
Series is convergent if L<1.
Thus b<1.
Therefore, for b<1, the series n=1bn is convergent.

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Jeffrey Jordon
Answered 2021-12-16 Author has 2047 answers

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