I need to solve for $a$ in the following: $$\lim_{n\to \infty}\dfrac{1^a+2^a+3^a+\cdots +n^a}{(n+1)^{a-1}\left[(na+1)+(na+2)+\cdots (na+n)\right]}=\dfrac{1}{60}$$
My Attempt:
$$\begin{aligned}\lim_{n\to \infty}\sum_{k=1}^{n}\dfrac{k^a}{(n+1)^{a-1}\sum_{k=1}^{n}(a+\frac{k}{n})\cdot\frac{1}{n}}&=\lim_{n\to \infty}\sum_{k=1}^{n}\dfrac{k^a}{(n+1)^{a-1}\int_{0}^{1}(a+x)\mathrm dx}\\ &=\lim_{n\to \infty}\dfrac{1}{a+1}\sum_{k=1}^{n}\dfrac{k^a}{(n+1)^{a-1}}\end{aligned}$$
I'm not sure how to proceed. Any hints and mistake pointers are appreciated. The answers provided in the answer key are $7$ and $-17/2$. Thanks