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.
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.
Hint
$ \frac{n^{k+1}}{n^k +k} =n \frac{1}{1+\frac{k}{n^k}}$
What happens when $n \to \infty$?
$\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 $