Dirichlet criterion (convergence of series)

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

A criterion for the convergence of the series $\sum_n a_n b_n$, where $a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet and published posthumously in [Di]. If a sequence of real numbers $a_n$ converges monotonically to zero, and the sequence of partial sums of the series $\sum_n b_n$ is bounded (the terms of this series may also be complex), then the series $\sum_n a_n b_n$ converges. The criterion is related to Dedekind's criterion.

References

[Di]	P.G.L. Dirichlet, "Démonstration d’un théorème d’Abel", J. de Math. (2) , 7 (1862) pp. 253–255