Limit $\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })}$ The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })}  $$
Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a }+2^{a }+\cdots+ n^{a} )}$
I am trying to use the integral as a limit of sum but the denominator is preventing me from applying.
Could anyone please give me a direction to move on.
Thank you for your help.

Hint: $1^a+2^a+\cdots+n^a\sim {n^{a+1}\over a+1}$.
