calculus - Find the limit of a series

$$ S(n, k) = \sum \limits_{s=1}^n \frac{\sin(n + \frac{s}{k})}{\sqrt[7]{nk^7 +sk^6}} $$

Prove that for every $n$ the limit $ G_n =\lim\limits_{k \to \infty}S(n,k)$ exists.
Show that sequence $\{G_n\}$ converges and find its limit.

I was thinking about Riemann sums but I don't even know where to start :(

Answer

For the first part, fix $n$. It is clear that each term of the summation given by $S(n,k)$ tends to $0$ as $k\to\infty$:$$-\frac1{\sqrt[7]{nk^7+sk^6}}\le\frac{\sin(n+s/k)}{\sqrt[7]{nk^7+sk^6}}\le\frac1{\sqrt[7]{nk^7+sk^6}}$$and clearly $\pm\dfrac1{\sqrt[7]{nk^7+sk^6}}\to0$.