I'm asked to find 2 limits. Since it's not real analysis, I assume the idea is to use generating functions - however, we didn't go through much of it, so I'm probably wrong about it.
Let $$ d \in\Bbb N, d \ge 1 $$ Let's define $$ S^d_n := \sum^{n}_{k=1}k^d $$ I'm asked to find the following limits:
$$ a_d := \lim_{n\to\infty} \frac{S^d_n} {n^{d+1} }$$
$$ \lim_{n\to\infty} \frac{S^d_n - (a_d) n^{d+1}}{n^{d} }$$
For the first one, ironically, I had the idea to use the Stolz–Cesàro theorem. I found that $$ a_d = \frac{1}{d+1} $$
For the second one, I figured that by using generating functions, I know that $$ \frac{1}{1-x} = \sum^{\infty}_{n=0} x^n $$ Deriving and then multiplying by x, we get that $$ \frac{x}{(1-x)^2} = \sum^{\infty}_{n=0} n x^n $$
I can do this until I reach $$n^d $$ However, I don't know how to find a "nice" way to to this; and why would this help? I can't let $$x=1$$ since the function is not defined there.
I'd appreciate any guidance or hints.
It seems that you are looking for the rate of convergence of the Riemmann sum for the function $x\mapsto x^d$. Look at this and the second limit will be $-\frac12$