This test criterion for convergent is due to Johann Lejeune Dirichlet (1805 - 1859).

Proposition: Dirichlet's Test

If the sequence of partial sums $(A_n)_{n\in\mathbb N}$ with $A_n:=\sum_{k=0}^n a_k$ of the infinite series $\sum_{k=0}^\infty a_k$ is bounded and the sequence $(b_k)_{k\in\mathbb N}$ is monotonic and1 convergent to zero $\lim_{k\to\infty} b_k=0,$ then the infinite series of products $\sum_{k=0}^\infty a_kb_k$ is convergent.

Proofs: 1

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  1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition


  1. If you want to apply this convergence criterion to complex-valued series $\sum_{k=0}^\infty a_kb_k$, you have to make sure that $(b_k)_{k\in\mathbb N}$ remains a real-valued sequence. Otherwise, the monotonicity criterion will lose its sense.