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.
