View source for Dedekind criterion (convergence of series)

{{MSC|40A05}}
{{TEX|done}}

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)|Dirichlet criterion (convergence of series)]].

====References====
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|valign="top"|{{Ref|Ru}}||  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)
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