TITLE: Show that $n^2\log\left(1+\frac{1}{n}\right)$ does not converge to $1$
QUESTION [7 upvotes]: How to show that $n^2\log\left(1+\dfrac{1}{n}\right)\to 1$ is false?

I have to show that $\left(1+\dfrac{1}{n}\right)^{n^2}$ doesn't tend to $e.$

REPLY [2 votes]: Another approach. Since
$$\left(1+\frac1n\right)^n\xrightarrow [n\to\infty]{}e>2.5$$
there exists $\,M\in\Bbb N\,$ s.t.
$$n>M\;\implies\; \left(1+\frac1n\right)^n>2$$
and thus, for $\,n>M$ :
$$\left(1+\frac1n\right)^{n^2}=\left[\left(1+\frac1n\right)^n\right]^n>2^n\xrightarrow [n\to\infty]{}\infty$$