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

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The series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305203.png" /> are complex numbers, converges if the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305204.png" /> converges absolutely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305205.png" />, and if the partial sums of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305206.png" /> are bounded.
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{{MSC|40A05}}
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{{TEX|done}}
  
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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.
  
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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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====Comments====
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\[
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The proof is based on the formula of summation by parts: Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305207.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305209.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d03052010.png" /> one has
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\sum_{n=p}^q a_n b_n = \sum_{n=p}^{q-1} B_n (a_q - a_{n+1}) + B_q a_q - B_{p-1} a_p \qquad \forall 1\leq p < q\, .
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\]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d03052011.png" /></td> </tr></table>
 
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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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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>
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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

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)
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=18271
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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