find $\lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{(k+1)^l}{k^{l+1}}, for: l\geq0$

Question

find $\lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{(k+1)^l}{k^{l+1}}, for: l\geq0$

We mostly studied Riemann sums so my main idea is to somehow express $f(\frac{k}{n})$ for some f using this sum.

I've found it's equal to $\lim_{n\rightarrow \infty}\sum_{k=1}^n(1+\frac{1}{k})^l\frac{1}{k}$ but i don't see how I can continue from here.

A hint on how to solve this with Riemann sums would be best :)

Answer 1

For any $l \ge 0$ $$\sum_{k=1}^n\left(1+\frac{1}{k}\right)^l\frac{1}{k} \gt \sum_{k=1}^n\frac{1}{k}$$ so the limit is $\infty$.