Abel test

Abel test: Consider a series $\sum_{n=1}^\infty a_n b_n$ of real or complex numbers. Suppose

1. the series $\sum_{n=1}^\infty a_n$ converges , and
2. the sequence $(b_n)$ is monotone and bounded.

Then the series $\sum_{n=1}^\infty a_n b_n$ converges.

Abel’s test can be viewed as complementary to the Dirichlet test : Dirichlet assumes bounded partial sums of $(a_n)$ and $b_n\to 0$, while Abel assumes convergence of $\sum a_n$ and only boundedness of $(b_n)$.