Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$

By considering:

$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = \frac 1 4$$

Determine if this is true: $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}} = \frac 1 {{m+1}}$$

If it is, prove it.

If it is not, evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{{m+1}}$.

Since this is tagged algebra-precalculus, I will refrain from using the Euler-Maclaurin Sum Formula (which is exact in the case of polynomials).

Let's first show that $$ \sum_{k=1}^nk^m\text{ is a polynomial of degree }m+1\text{ in }n\text{ whose lead coefficient is }\frac{1}{m+1}\tag{1} $$

Since $\displaystyle\sum_{k=1}^n1=n$, we have shown that $(1)$ is true for $m=0$.

Suppose that $(1)$ is true for all $m<M$, then $$ k^{M+1}-(k-1)^{M+1}=\sum_{m=0}^{M}\binom{M+1}{m}(-1)^{M-m}k^m\tag{2} $$ If we sum $(2)$ for $k$ from $1$ to $n$, the telescoping sum on the left yields $$ n^{M+1}=\sum_{m=0}^{M}\binom{M+1}{m}(-1)^{M-m}\left(\sum_{k=1}^nk^m\right)\tag{3} $$ Isolating the $m=M$ term from the right-hand side of $(3)$ gives us $$ \sum_{k=1}^nk^M=\frac{1}{M+1}n^{M+1}-\frac{1}{M+1}\sum_{m=0}^{M-1}\binom{M+1}{m}(-1)^{M-m}\left(\sum_{k=1}^nk^m\right)\tag{4} $$ Since $(1)$ holds for each $m<M$, each $\displaystyle\left(\sum_{k=1}^nk^m\right)$ on the right-hand side of $(4)$ is a polynomial in $n$ of degree $m+1<M+1$. Thus, because of the $n^{M+1}$, the right-hand side of $(4)$ is a polynomial in $n$ of degree $M+1$ whose lead coefficient is $\dfrac{1}{M+1}$.

Thus, we have shown $(1)$ for all $m\ge0$.

Consider $$ \frac{1}{n^{m+1}}\sum_{k=1}^nk^m\tag{5} $$ The sum in $(5)$ is a polynomial in $n$ of degree $m+1$ whose lead coefficient is $\dfrac{1}{m+1}$. Therefore, $(5)$ becomes $$ \frac{1}{m+1}+O\left(\frac1n\right)\tag{6} $$ Thus, we get that $$ \lim_{n\to\infty}\;\frac{1}{n^{m+1}}\sum_{k=1}^nk^m=\frac{1}{m+1}\tag{7} $$

This follows easily from a comparison with an integral.

Since $$\int_0^{n} k^m dk \leq \sum_{k=1}^n k^m \leq \int_1^{n+1} k^m dk,$$ we have $$\frac{n^{m+1}}{m+1}\leq \sum_{k=1}^n k^m \leq \frac{(n+1)^{m+1}}{m+1}-\frac{1}{m+1}.$$ Divide by $n^{m+1}$ to get $$\frac{1}{m+1} \leq \frac{1}{n^{m+1}} \sum_{k=1}^n k^m \leq \frac{1}{m+1} \cdot \frac{(n+1)^{m+1}}{n^{m+1}} - \frac{1}{m+1} \cdot \frac{1}{n^{m+1}}.$$ Taking $n \rightarrow \infty$ gives the desired result.