Monotone decay to zero times bounded partial sums gives convergence.Suggest a slogan
Let \((a_n)_{n\ge1}\) be a sequence of real numbers and \((b_n)_{n\ge1}\) a sequence of complex numbers such that
\(a_{n+1} \leq a_n\) for all \(n\) and \(a_n \to 0\), and
there exists \(M\ge0\) such that \(\abs{\sum_{n=1}^N b_n} \leq M\) for all \(N\ge1\).
Then the series \(\sum_{n=1}^\infty a_n b_n\) converges.
State Dirichlet's test for convergence of a series.