Strategy for Series — Question 8

*Question 8 Determine whether $\displaystyle\sum_{n=1}^{\infty}\left(1+\frac1n\right)^{-n^2}$ converges. Explain why the Root Test is the natural first choice.

*Question 8 – Solution

Let $a_n=(1+1/n)^{-n^2}$. Taking an $n$th root cancels one power of $n$: $$\sqrt[n]{a_n}=\left(1+\frac1n\right)^{-n}.$$

Since $(1+1/n)^n\to e$, $$L=\lim_{n\to\infty}\sqrt[n]{a_n}=e^{-1}<1.$$

The series converges by the Root Test. This method is especially efficient because the summand is already a varying base raised to a multiple of $n$; the root exposes its limiting geometric rate directly.