# Series Convergence by Comparison

1. $$
\sum_{n=m}^\infty \frac{C}{n^{1+\epsilon}}
\leq \int_{m-1}^\infty \frac{C}{x^{1+\epsilon}} \operatorname{d}\!
= \left[-\frac{C}{\epsilon x^\epsilon}\right]_{m-1}^\infty
= \frac{C}{\epsilon (m-1)^\epsilon} - 0
< \infty $$
2. If $\lvert a_n \rvert$ is absolutely convergent, $\sum_{n=m}^\infty \frac{1}{n}$ is convergent by comparison, but this is contradicts with the fact that $\sum_{n=m}^\infty \frac{1}{n}$ is divergent.