# Difference between revisions of "Dedekind criterion (convergence of series)"

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+ | 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 | |

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− | The | + | \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\, . |

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A related convergence criterion is the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]]. | A related convergence criterion is the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]]. | ||

====References==== | ====References==== | ||

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

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## Latest revision as of 20:29, 9 December 2013

2010 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) |

**How to Cite This Entry:**

Dedekind criterion (convergence of series).

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dedekind_criterion_(convergence_of_series)&oldid=30913