If $u_n=\frac{1}{\left(1+\frac{1}{n}\right)^{n^2}}$, the series $\sum u_n$ is convergent, if $\lim _

Question

If $u_n=\frac{1}{\left(1+\frac{1}{n}\right)^{n^2}}$, the series $\sum u_n$ is convergent, if $\lim _{n \rightarrow \infty}\left(u_n^{\frac{1}{n}}\right)$

Professor PC Thomas Classes · Answer

By Cauchy's root test, $\lim_{n 	o \infty} (u_n)^{1/n} = \lim (1 + 1/n)^{-n} = 1/e < 1$, dictating convergence.