Root Test — Question 6

*Question 6 Analyze $\displaystyle\sum_{n=1}^{\infty}\left(\frac{n+1}{2n}\right)^n$ using the Root Test. State the root limit and the convergence classification.

*Question 6 – Solution

$$\sqrt[n]{\left|\left(\frac{n+1}{2n}\right)^n\right|}=\frac{n+1}{2n}=\frac12+\frac1{2n}.$$

$$L=\lim_{n\to\infty}\left(\frac12+\frac1{2n}\right)=\frac12<1.$$

The series converges absolutely by the Root Test. Although the base varies with $n$, it approaches $1/2$, so the terms eventually have geometric-scale decay. For example, once $n\ge5$, the root is at most $3/5$, so $a_n\le(3/5)^n$.