Root Test — Question 9

Question 9 – Solution

Step 1: Rewrite the term.

We have $a_n=n^{-n}=(1/n)^n$.

Step 2: Take the $n$th root.

$$\sqrt[n]{|a_n|}=\sqrt[n]{\left(\frac1n\right)^n}=\frac1n.$$ Therefore $$L=\lim_{n\to\infty}\frac1n=0<1.$$

Conclusion.

The series converges absolutely. Rather than complicating the calculation, the exponent $n$ is exactly canceled by the $n$th root. Moreover, for any $0<r<1$, eventually $1/n<r$, so $n^{-n}<r^n$ and the tail is dominated by a geometric series.