Find all possible values of $p$ such that $\sum_{k=2}^∞ ln(1+\frac{(- 1)^k}{k^p}) $ Converges.

Asked Jan 21 '22 at 14:15

Active Jan 21 '22 at 14:21

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This is a calculus homework question:

Find all possible values of $p$ such that: $$\sum_{k=2}^\infty \ln\left(1+\frac{(-1)^k}{k^p}\right)$$Converges.

I tried to use comprasion test, But all candidates failed to achieve the desired result.

I already spent alot of time on it, so any assistance will be useful.

edited Jan 21 '22 at 14:21

FShrike

asked Jan 21 '22 at 14:15

MStudent

1

Just being curious : what is a lot of time ? I am serious since it is a very relative concept – Claude Leibovici Jan 21 '22 at 14:24
@ClaudeLeibovici Around 5 hours. – MStudent Jan 21 '22 at 14:26
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If you meant $p \in \Bbb{R}$ then what do you get for $p > 1$? For $p\in (1/2,1]$ ? For $p=1/2$ ? For $p < 1/2$ grouping the terms by two will help – reuns Jan 21 '22 at 14:27
Use this estimate: https://math.stackexchange.com/q/2267122/42969. – Martin R Jan 21 '22 at 14:28
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Or see here: https://math.stackexchange.com/q/597060/42969, or here: https://math.stackexchange.com/q/3239056/42969, or here: https://math.stackexchange.com/q/541672/42969 – all found with Approach0 – Martin R Jan 21 '22 at 14:29

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