Root Test — Question 5

*Question 5 Determine whether $$\sum_{n=2}^{\infty}\left(\frac{\log n}{n}\right)^n$$ converges or diverges. Compute the Root Test limit and interpret what a limit of zero implies.

*Question 5 – Solution

Since $n\ge2$, $$\sqrt[n]{\left|\left(\frac{\log n}{n}\right)^n\right|}=\frac{\log n}{n}.$$

By l’Hopital’s Rule for the corresponding continuous quotient, $$\lim_{x\to\infty}\frac{\log x}{x}=\lim_{x\to\infty}\frac{1/x}{1}=0.$$ Hence $L=0<1$.

The series converges absolutely. A root limit of zero means that for every fixed $0<r<1$, the terms eventually satisfy $a_n\le r^n$; thus the tail is eventually dominated by a geometric series with any chosen ratio $r$.