I am having difficulty finding if this series converges or d

Teddy Dillard 2021-12-25 Answered
I am having difficulty finding if this series converges or diverges:
n=0(2)n+1nn1
I am unsure of which test to use. At first, I thought I should use alternating series test, but I am unable to manipulate the numerator to (1)n+1. I thought I may be able to use root test, but I have no clue how to manipulate the series to do that.
What series test would I use, and how?
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Expert Answer

Ethan Sanders
Answered 2021-12-26 Author has 35 answers
Alternating series test
Note that
n=0(2)n+1nn1=n=0(1)n+12n+1nn1
By the alternating series test this converges if
1. an>0: we can easily this is satisfied
2. an is monotic: an is decreasing monotonically
3. limnan=0 I will leave this one to you.
Comparison test
|n=0(2)n+1nn1|n=02n+1nn1n=03(2n)n+1+n=4(2n)n+1
The last series is a geometric series and therefore converges.
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Virginia Palmer
Answered 2021-12-27 Author has 27 answers
Hint: what can you say about 2n1nn1?
Can you show that 2n1nn11n2 for nN?
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user_27qwe
Answered 2021-12-30 Author has 208 answers

Making the problem more general, let
S=n=0an with an=xn+1nn1
log(an)=(n+1)log(x)(n1)log(n)
log(an+1)log(an)=(n1)log(n)nlog(n+1)+log(x)
Using Taylor for large values of n
log(an+1)log(an)=log(xne)+12n+O(1n2)
an+1an=elog(an+1)log(an)xne0

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