Comparison Test

Comparison test: Let $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ be series with $a_n\ge 0$ and $b_n\ge 0$ for all $n$.

• If $0 \le a_n \le b_n$ for all sufficiently large $n$ and $\sum b_n$ is a convergent series , then $\sum a_n$ is convergent.
• If $0 \le a_n \le b_n$ for all sufficiently large $n$ and $\sum a_n$ is a divergent series , then $\sum b_n$ diverges.

This test reduces many convergence questions to comparisons with standard benchmark series, and it is complemented by the limit comparison test when direct inequalities are hard to establish.