Prove $(1+1/n)^{n(n+1)}$ diverges to infinity (as $n\to \infty$)

Question

Prove $(1+1/n)^{(n(n+1))}$ diverges to infinity (as $n\to\infty$) .

I am unsure how to do so using $a > K$ argument.

Thank you :)

Answer 1

$(1+1/n)^n\to e$ as $n\to \infty$, so in particular $\exists N$ so that for all $n>N$, $(1+1/n)^n\geq 2$. This means that $$(1+1/n)^{n(n+1)}\geq 2^{n+1}.$$ Do you know how to finish it off from here?

Answer 2

HINT: it is $$\left(\left(1+\frac{1}{n}\right)^n\right)^{n+1}$$

Answer 3

Simply note that $(1+\frac{1}{n})^n)$ tends to $e$ as $n$ goes to infinity, hence the expression is approximate to $e^{n+1}$ as $n$ becomes sufficiently large. The rest of the proof can be done through an epsilon-delta argument, but it is sufficient to note that this exponential clearly diverges as $n$ goes to infinity (if your proof is not required to be rigorous).

Answer 4

Using the simple binomial expansion, $$\left(1+\frac{1}{n}\right)^{n(n+1)}=\sum_{k=0}^{n(n+1)}\,\binom{n(n+1)}{k}\,\frac{1}{n^k}\geq 1+\binom{n(n+1)}{1}\,\frac{1}{n}=n+2\,.$$