How to determine the limit of this sequence? I was wondering how to determine the limit of $ (n^p - (\frac{n^2}{n+1})^p)_{n\in \mathbb{N}}$ with $p>0$, as $n \to \infty$?
For example, when $p=1$, the sequence is $ (\frac{n}{n+1})_{n\in \mathbb{N}}$, so its limit is $1$.
But I am not sure how to decide it when $p \neq 1$. Thanks in advance!

We can write 
$$a_n=n^p\left(1-\left(1-\frac 1{n+1}\right)^p\right)=n^p\int_{1-\frac 1{n+1}}^
1pt^{p-1}dt.$$
If $p>1$, the map $t\mapsto t^{p-1}$ is increasing so $a_n\geq n^pp\left(1-\frac 1{n+1}\right)^{p-1}\frac 1{n+1}$ which converges to $+\infty$ and if $0<p<1$ then $a_n\leq n^pp\left(1-\frac 1{n+1}\right)^{p-1}\frac 1{n+1}$ which converges to $0$.
