Question

Hello!
What is the easiest way to study the convergence or divergence of this serie?

\[\sum _{n=1}^{\infty \:}\left(1+\frac{1}{n}\right)^{n^2}\]

Answer 1

@zepp Cauchy's root test

Answer 2

\[(1+1/n)^{n^2}=((1+1/n)^{n})^{n}\]

Answer 3

Oh! I didn't see that, I'll work from there, thanks!

Answer 4

Hints, the limit is e . now apply ratio test. and tell me the answer

Answer 5

e>1

Answer 6

Yeah, by the root test test, \(\lim_{x \rightarrow \infty}\ |{a_{n}^{\frac{1}{n}}}|=e>1\) thus the series diverges

Answer 7

right

Answer 8

Thank you!