Ratio Test — Question 10

*Question 10 Use the Ratio Test to analyze $\displaystyle\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^2}$. Use logarithms and a standard logarithmic expansion to evaluate the ratio limit.

*Question 10 – Solution

Let $a_n=(n/(n+1))^{n^2}$. Then $$\log a_n=-n^2\log\left(1+\frac1n\right).$$ Using $\log(1+t)=t-t^2/2+O(t^3)$, $$\log a_n=-n^2\left(\frac1n-\frac1{2n^2}+O(n^{-3})\right)=-n+\frac12+O(n^{-1}).$$

$$\log\left(\frac{a_{n+1}}{a_n}\right)=\log a_{n+1}-\log a_n=-1+O(n^{-1})\longrightarrow-1.$$ Exponentiating gives $L=e^{-1}<1$.

The series converges. More specifically, $a_n\sim e^{1/2}e^{-n}$, so its terms have geometric-scale decay.