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How to prove con/di vergence of $\sum \limits_{k=1}^\infty (\frac{k^3}{-1-k^4})=\sum \limits_{k=1}^\infty -(\frac{k^3}{k^4+1})$
My problem is that I don't know how to do that I tried some different methodes but it doesn't work.Does anyone have an idea?Can I prove it by the p-test although there is a (-1)?

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up vote 1 down vote accepted

The sum in your question diverges. You can use the limit comparison test with the harmonic series to show that it converges iff the harmonic series converges.

Another way: $$\sum_{k=1}^{\infty}\frac{1}{2k}\leq\sum_{k=1}^{\infty}\frac{1}{k+\frac{1}{k}}=\sum_{k=1}^{\infty}\frac{k^3}{k^4+1}$$

This follows from the fact that $\forall k\in Z^+[2k\geq k+\frac{1}{k}]$

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Thank you that was really helpful.My fault was that I have tried to do it by a majorant which diverges... – phil Jan 11 '13 at 6:49

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