Does the infinite series converge with n^k in the numerator and the polynomial n^(k+1) + n^k + n^(k-1) + ... 1 as k goes to infinite?

Question

I know that I am asking this question the wrong way, but hopefully one of you understands what I am getting at. Does it converge? $$ \lim_{k\rightarrow \infty } \sum_{n = 0}^{\infty} \frac{n^k}{n^{k+1}+n^{k}+... +n +1} $$

Answer 1

No. To see it, use equivalents: $$n^{k+1}+n^k\dots+n+1=\frac{n^{k+2}-1}{n-1}\sim_\infty\frac{n^{k+2}}{n}=n^{k+1},$$ hence $\;u_n\sim_\infty\dfrac{n^k}{n^{k+1}}=\dfrac 1n$.