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!

Given the Binomial theorem, we have (see Landau notations)
$$\begin{array}{} (n+1)^p-n^p=\Theta(n^{p-1}) & \implies 1-\left(\frac{n}{n+1}\right)^p=\Theta \left( \frac{1}{n} \right) \\ & \implies n^p - \left(\frac{n^2}{n+1}\right)^p=\Theta(n^{p-1}) \end{array}$$
Thus the limit is $0$ for $p<1$, is $1$ for $p=1$ (already computed), and blows up to $\infty$ for $p>1$.
