Does $\sum_{n=1}^\infty(1- \sqrt[n]{n})^n$ converge absolutely?

Question

I want to determine if $\sum_{n=1}^\infty(1- \sqrt[n]{n})^n$ converges absolutely or conditionally.

Note that $\lim_{n \to \infty} \sqrt[n]{a_n}=\lim_{n \to \infty}(1- \sqrt[n]{n})=0<1$ so the sequence converges.

Now I need to find it for $\sum_{n=1}^\infty|(1- \sqrt[n]{n})^n|$, but I don't know what I should do.

Any suggestions?

Thanks!

Answer 1

For each $n\in\Bbb N$,$$\left|\left(1-\sqrt[n]n\right)^n\right|=\left(\sqrt[n]n-1\right)^n.$$Since$$\lim_{n\to\infty}\sqrt[n]{\left(\sqrt[n]n-1\right)^n}=\lim_{n\to\infty}\sqrt[n]n-1=0<1,$$the series $\displaystyle\sum_{n=1}^\infty\left|\left(1-\sqrt[n]n\right)^n\right|$ converges; in other words, the series $\displaystyle\sum_{n=1}^\infty\left(1-\sqrt[n]n\right)^n$ is absolutely convergent.