(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine whether the series [tex]\sum_{n=2}^{\infty}a_n[/tex] is absolutely,conditionally convergent or divergent

[tex]a_n=\frac{(-1)^n}{\sqrt{n}(\frac{2n}{n+1})^\pi}[/tex]

3. The attempt at a solution

from Abel's test.[tex]c_n=\frac{(-1)^n}{\sqrt{n}}[/tex]is convergent.and

[tex]b_n=(\frac{2n}{n+1})^\pi}=\frac{2^{\pi}}{(1+\frac{1}{n})^{\pi}}=\frac{2^{\pi}}{1+\frac{\pi}{n}+o(\frac{1}{n^2})}[/tex].Which has limit [tex]2^{\pi}[/tex].So a_n is convergent.

[tex]|a_n|=\frac{2^{\pi}}{\sqrt{n}(1+\frac{1}{n})^{\pi}}=\frac{2^{\pi}}{\sqrt{n}+\frac{\pi}{\sqrt{n}}+O(\frac{1}{\sqrt{n}n})}[/tex]

I don't know exactly but it seems to me that the last equation is divergent.So a_n is conditionally convergent.

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# Homework Help: Determine whether the series is convergent

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