Real Analysis: Limit Proof with help of ratio convergence property

Question

The thing is to, using the property above, prove the expression below. First tried to obtain the difference between n and n+1 terms, then factored out n+1 ^ p and rearanged the sequence. I've represented the latest with 1/ [ n+1 - ( 1 - 1/(n+1) ) ^ p+1 n+1 ] I thought about using bernoulli inequality, but am not sure Need some ideas or tips on how to finish the proof please Thanks

Answer 1

You can evaluate the limit in a different way if you view it as a Riemann integral.

Notice that $\frac{1^p + 2^p + ... + n^p}{n^{p+1}} = \frac{1}{n} \sum_{j=1}^n (j/n)^p \to \int_0^1 x^p dx = \frac{1}{p+1}$ as $n \to \infty$.

Answer 2

Finally found out how to do this thing

Thought it could be useful to others