Prove this inequality by math induction

Question

$$\sum \limits_{k=1}^{n-1} k^p < \frac{ n^{p+1}}{p+1} < \sum\limits_{k=1}^n k^p $$

I know how to prove it by using Riemann Sum, but it I was thinking if there is anyway to do it by mathematical induction?

Answer 1

You just have to prove that: $$\frac{n^{p+1}}{p+1}+n^p < \frac{(n+1)^{p+1}}{p+1}\tag{1}$$ and: $$\frac{n^{p+1}}{p+1}+(n+1)^p > \frac{(n+1)^{p+1}}{p+1}\tag{2}$$ or: $$(n+1)^{p+1}>n^{p+1}+(p+1)n^p\tag{1*}$$ $$n^{p+1}+p(n+1)^p > 0\tag{2*}$$ where $(1*)$ follows from the binomial theorem and $(2*)$ is trivial.