Show if the series $\sum _{n=1}^{\infty} \frac {3}{9n+1}$ converges or diverges.

Question

Show if the series $\sum _{n=1}^{\infty} \frac {3}{9n+1}$ converges or diverges.

The solution says that by comparison test with harmonic series this series diverges. But, $\frac {1}{n}$ is larger than $\frac {1}{9n+1}$. That is, the divergence of harmonic series does not tell us whether $\sum _{n=1}^{\infty} \frac {3}{9n+1}$ converges or not.

Am I correct? Could you give me the alternative to solve this problem?

Answer 1

\begin{align} \sum_{n=1}^{\infty}\frac{3}{9n+1} &= \frac{1}{3}\sum_{n=1}^{\infty}\frac{1}{n+1/9}\\ &=\frac{1}{3}\sum_{n=2}^{\infty}\frac{1}{n-8/9} \gt\frac{1}{3}\sum_{n=2}^{\infty}\frac{1}{n}\\ \text{And the last term diverges by harmonic series} \end{align}