Does the attached series converge or diverge? Note: it is a complex series and not a real series. I've been told the limit comparison test is not a valid approach to solving this problem. sum_{n=1}^inftyfrac{1}{n-i^n}

Clifland 2021-01-04 Answered
Does the attached series converge or diverge? Note: it is a complex series and not a real series. Ive
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Expert Answer

Laaibah Pitt
Answered 2021-01-05 Author has 98 answers

The given series is
n=11nin
Let us solve the given problem by the help of the Ratio test.
The Ratio test is,
For a series an, If limn|an+1an|=L
Then, the series is convergent if L<1
the series is divergent if L>1
And if L=1, we have no result or we can say the test fails.
Here an=1nin
So, limn|an+1an|=limn|nin(n+1)in+1|()
=limn|1inlni1in+1lni|()
=limn|in(lni)2in+1(lni)2|
=1i
=1i(ii)
=-i,(as i2=1)
Therefore, L=i<1
So, the given series is convergent.
ANSWER: The given series is convergent.

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Jeffrey Jordon
Answered 2021-12-16 Author has 2313 answers

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