(Dis-)proving the series $\sum\limits_n\left( 1+ \frac{1}{n} \right)^n$ converges I am trying to prove that the series: $$\sum^\infty_{n=1}\left( 1+ \frac{1}{n} \right)^n$$ converges. Now I know that $$\lim_{n\rightarrow\infty} \left( 1+ \frac{1}{n} \right)^n=e$$ But how can I use that knowledge to prove the convergance ? Intuitively I would say that the series diverges since it doesn't approach zero but how can I formally prove this?

A necessary condition for the series $$ \sum a_j $$ to converge is $a_j\to0$ as $j\to\infty$. If you can show that $a_j\
ot\to0$ as $j\to\infty$, then this implies that the series diverges. See here for more details.