I want to show this inequality $$\frac{m^{n+1}}{n+1} < 1^n + 2^n + \dots + m^n <\frac{\left(m+1\right)^{n+1}}{n+1}$$ Where $ m $ and $ n $ are positive integers.
I have a hint that suggests I use the inequality $$(n+1)(t-1)<t^{n+1} - 1 <(n+1) t^{n} (t-1) $$ Where $ t>1 $ is real and $ n>1 $ is an integers. Specifically I'm having trouble with the sum $$ 1^n + 2^n + \ldots + m^n $$
For the first of the two inequalities, consider $f(x)=x^n$ on the interval $[0,m]$ and note that the sum is the right hand Riemann sum when $[0,m]$ is partitioned into $m$ equal subintervals. Since $f$ is strictly increasing, the right hand Riemann sum is greater than $\int_0^m x^n=m^{n+1}/(n+1).$
There may be a similar proof for the other inequality. I agree this method may not be how you want to prove it, since it doesn't use your given hint.
For the second inequality, again consider $f(x)=x^n,$ but this time on the interval $[0,m+1].$ This time the given sum is the left endpoint Riemann sum when the interval $[0,m+1]$ is partitioned into $m+1$ equal parts, and since the sum is a left endpoint Riemann sum it is less than $\int_0^{m+1} x^n=(m+1)^{n+1}/(n+1).$
