Find the region of convergence and absolute convergence of the series. sum_{n=1}^inftyfrac{ln^n x}{n}

Marvin Mccormick 2020-11-27 Answered
Find the region of convergence and absolute convergence of the series.
n=1lnnxn
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Expert Answer

Brighton
Answered 2020-11-28 Author has 103 answers

Given the series, n=1lnnxn
Here, |an|=|(lnx)nn|
To find the region of convergence and absolute convergence of the given series.
By ratio test, if limnan+1an<1, the series converges.
Therefore, the absolute convergence of the series implies,
limn|an+1an|=limn|(lnx)n+1n+1×n(lnx)n|
=limn|nlnxn+1|
=limn|lnx1+1n|
=|lnx|
So, the series absolutely converges if, |lnx|<1
Therefore, the region of convergence is, |lnx|<1

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Jeffrey Jordon
Answered 2021-12-27 Author has 2087 answers

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