Root Test — Question 1

*Question 1 Consider $$\sum_{n=1}^{\infty}\frac{(2n+1)^n}{5^n n^n}.$$

1. Rewrite the summand as a single quantity raised to the $n$th power.

2. Apply the Root Test and classify the series.

*Question 1 – Solution

Let $$a_n=\frac{(2n+1)^n}{5^n n^n}=\left(\frac{2n+1}{5n}\right)^n=\left(\frac25+\frac1{5n}\right)^n.$$

Since $a_n>0$, $$\sqrt[n]{|a_n|}=\frac{2n+1}{5n}=\frac25+\frac1{5n}.$$ Therefore $\displaystyle L=\lim_{n\to\infty}\sqrt[n]{|a_n|}=2/5<1$.

The series converges absolutely by the Root Test. The slowly varying $1/(5n)$ correction disappears in the limit, leaving geometric-scale decay with factor $2/5$.