Using the fact that for all integers $\displaystyle n\geqslant1$ and real numbers $\displaystyle t>1$

$\displaystyle (n+1)t^n(t-1)>t^{n+1}-1>(n+1)(t-1),$

prove that for all positive integers m and n,

$\displaystyle \frac{m^{n+1}}{n+1}<1^n+2^n+...+m^n<\left(1+\frac{ 1}{m}\right)^{n+1}\frac{m^{n+1}}{n+1}.$

I think it has something to do with summing the first inequalities over t, but I've been going round and round in circles with various summations, and can't seem to turn it into the second inequality

. Can anyone give me a nudge in the right direction?