Calculus (3rd Edition)

We have to find $\rho$ $$\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty}\left|\frac{1/2^{n+1}}{1/2^n}\right|=\frac{1}{2}\lt1$$ Hence the Ratio Test is conclusive for $\frac{1}{2^n}$. $$\rho= \lim _{n \rightarrow \infty}\left|\frac{1/(n+1)}{1/n}\right|= \lim _{n \rightarrow \infty}\frac{n}{n+1}=1$$ Hence the Ratio Test is inconclusive for $\frac{1}{n}$.