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Problem: Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$, where $k$ is a positive constant.

I'm stumped. I've tried to apply several different convergence tests, but still can't figure this one out.

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Ugh, I mistyped the series. Should I ask a new question? It should've been $\sum_{n=0}^\infty \frac{n^{k-1}}{n^k+k}$. – Damir Aug 25 '12 at 8:55
Use the theorem "If a series $\sum_{n=0}^{\infty} a_n $ converges, then $\lim_{n\rightarrow \infty} a_n =0$". – Mhenni Benghorbal Aug 25 '12 at 11:34
up vote 3 down vote accepted


$$ \frac{n^{k+1}}{n^k +k} =n \frac{1}{1+\frac{k}{n^k}}$$

What happens when $n \to \infty$?

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$$\frac{n^{k+1}}{n^k+k}\geq\frac{n^{k+1}}{2n^k}=\frac{1}{2}n\xrightarrow [n\to\infty]{}\infty\neq 0 $$

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