2
$\begingroup$

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.

  • 0
    Ugh, I mistyped the series. Should I ask a new question? It should've been $\sum_{n=0}^\infty \frac{n^{k-1}}{n^k+k}$.2012-08-25
  • 0
    Use the theorem "If a series $\sum_{n=0}^{\infty} a_n $ converges, then $\lim_{n\rightarrow \infty} a_n =0$".2012-08-25

2 Answers 2

3

Hint

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

What happens when $n \to \infty$?

3

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