Inequality proof in Tom Apostol's calculus book (Thm 1.15)

Question

I'm reading Tom Apostol's calculus and I'm a bit stuck on proving an inequality which is part of the proof of theorem 1.15 (the integral of $x^p$).

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

Now the author says this can be proven by induction, but I have been trying for a few days without success! I'm sure it's pretty straightforward but for some reason I cannot get to it.

I would really be grateful if someone could help me finding the way out of this!

Thanks Fabrizio

Answer 1

Hint. In the inductive step you have to show that $$\frac{n^{p+1}}{p+1}+n^p\leq \frac{(n+1)^{p+1}}{p+1}$$ that is $$(p+1)n^p\leq (n+1)^{p+1}-n^{p+1},$$ or $$\frac{p+1}{n}\leq \left(1+\frac{1}{n}\right)^{p+1}-1,$$ then use Bernoulli's inequality.