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Finding whether the series $\displaystyle \sum^{\infty}_{k=2}\frac{(-1)^k 4^{k}}{k^{10}}$ is absolutely convergent, conditionally convergent or divergent

What i try :: For absolutely convergent/Divergent.

Let $\displaystyle a_{k}=\frac{4^k}{k^{10}}$ and $\displaystyle a_{k+1}=\frac{4^{k+1}}{(k+1)^{10}}$.

Then using ratio test $$\lim_{k\rightarrow \infty}\bigg|\frac{a_{k+1}}{a_{k}}\bigg|=4>1$$

So the series is Diverges.

But i did not understand How can i prove series is conditionally converges or not. Help me please. Thanks

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1 Answer 1

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Since $$\lim_{k \rightarrow +\infty} \left|\frac{(-1)^k4^k}{k^{10}}\right| = +\infty$$

the series diverges.

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