Root Test — Question 7

*Question 7 Use the Root Test to determine whether $$\sum_{n=2}^{\infty}\left(1-\frac1{\sqrt n}\right)^{n^2}$$ converges. Use logarithms to evaluate the limit of the $n$th root.

*Question 7 – Solution

$$r_n=\sqrt[n]{a_n}=\left(1-\frac1{\sqrt n}\right)^n.$$

Since $\log(1-u)\le-u$ for $0<u<1$, $$\log r_n=n\log\left(1-\frac1{\sqrt n}\right)\le -\sqrt n\longrightarrow-\infty.$$ Thus $0\le r_n\le e^{-\sqrt n}\to0$. (The expansion $\log(1-u)=-u-u^2/2+O(u^3)$ gives the sharper relation $\log r_n=-\sqrt n-1/2+O(n^{-1/2})$.)

The root limit is $L=0<1$, so the series converges absolutely and does so faster than any fixed geometric rate.