Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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)?

share|improve this question

1 Answer 1

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 $\forall k\in Z^+[2k\geq k+\frac{1}{k}]$

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.