Q:

Proving the limit of the following sequence

Prove that the following sequence is convergent and calculate it's limit:
$ a_n = (1+\frac{1}{n^2+2n} ) ^{n^2} $
I had this question in an exam today and failed miserably. I tried using squeeze theorem, sub-sequences, but I could not find a way.
Intuitively the limit is e, how to solve this?
Thanks in advance.

A:

An idea:
$$\left(1+\frac1{n^2+2n}\right)^{n^2}=\left(1+\frac1{n^2+2n}\right)^{n^2+2n}\left(1+\frac1{n^2+2n}\right)^{-2n}\xrightarrow[n\to\infty]{}e\cdot1=e$$

