Root Test — Question 1

Question 1 – Solution

Step 1: Rewrite the term.

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

Step 2: Take the $n$th root.

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

Conclusion.

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