Dirichlet's test

Let \((a_n)_{n\ge1}\) be a sequence of real numbers and \((b_n)_{n\ge1}\) a sequence of complex numbers such that

• \((a_n)\) is nonincreasing and \(a_n\to0\), and

• there exists \(M\ge0\) such that \(\abs{\sum_{n=1}^N b_n} \leq M\) for every \(N\ge1\).

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