## Calculus (3rd Edition)

Yes, the series $\Sigma_{n=1}^{\infty}a_n^{-1}$ converges.
Let $b_n=a_n^{-1}$; then applying the ratio test, we have $$\rho=\lim _{n \rightarrow \infty}\left|\frac{b_{n+1}}{b_{n}}\right|=\lim _{n \rightarrow \infty} \left|\frac{a_{n+1}^{-1}}{a_{n}^{-1}}\right|\\ =\lim _{n \rightarrow \infty} \left|\frac{a_{n } }{a_{n+1}}\right|=\frac{1}{4}\lt1$$ Hence, the series $\Sigma_{n=1}^{\infty}a_n^{-1}$ converges.