Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$

Question

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.

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 08:55
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

score 3 · Accepted Answer · answered Aug 25 '12 at 01:33

3

Hint

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

What happens when $n \to \infty$?

answered Aug 25 '12 at 01:33

N. S.

score 3 · Answer 2 · answered Aug 25 '12 at 01:33

3

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

answered Aug 25 '12 at 01:33

DonAntonio

2 Answers2