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Difference between revisions of "Du Bois-Reymond criterion (convergence of series)"

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A series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034040/d0340401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034040/d0340402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034040/d0340403.png" /> are complex numbers, converges if the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034040/d0340404.png" /> converges and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034040/d0340405.png" /> converges absolutely. Established by P. du Bois-Reymond.
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A criterion for the convergence of series of complex numbers, established by P. du Bois-Reymond.
  
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If the series of complex numbers $\sum_n b_n$ converges and the series of complex numbers $\sum_n (a_n-a_{n+1})$ converges absolutely, then the series $\sum_n a_n b_n$ converges.
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====Comments====
 
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====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964) pp. 324 (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR></table>
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|valign="top"|{{Ref|Kn}}|| K. Knopp,  "Theorie und   Anwendung der unendlichen Reihen" , Springer  (1964)  (English   translation: Blackie, 1951 &amp; Dover, reprint, 1990)
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Latest revision as of 13:16, 10 December 2013

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

A criterion for the convergence of series of complex numbers, established by P. du Bois-Reymond.

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

References

[Kn] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Du Bois-Reymond criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Du_Bois-Reymond_criterion_(convergence_of_series)&oldid=30930
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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