Dedekind 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 $a_n, b_n$ are complex numbers. If the series $\sum_n (a_n - a_{n+1})$ converges absolutely and the partial sums of the series $\sum_n b_n$ are bounded, then $\sum_n a_n b_n$ converges.

The criterion is based on the formula of summation by parts (a discrete analog of the Integration by parts): if we set $B_n = \sum_{k=1}^n b_k$ (with the convention that $B_0 = 0$), then \[ \sum_{n=p}^q a_n b_n = \sum_{n=p}^{q-1} B_n (a_q - a_{n+1}) + B_q a_q - B_{p-1} a_p \qquad \forall 1\leq p < q\, . \] A related convergence criterion is the Dirichlet criterion (convergence of series).

References

[Ru]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976)