Dirichlet test

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

denote the partial sums of $\sum a_n$. If

1. the sequence $(A_n)$ is bounded, and
2. $(b_n)$ is monotone and $b_n\to 0$,

then $\sum_{n=1}^\infty a_n b_n$ converges .

The alternating series test is a special case (take $a_n=(-1)^{n-1}$), and Dirichlet’s test is closely paired with the Abel test .