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A [[Series|series]] whose terms are also series:
 
A [[Series|series]] whose terms are also series:
  
<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/r/r081/r081300/r0813001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\sum_{n=1}^\infty\left(\sum_{m=1}^\infty u_{mn}\right).\label{1}\tag{1}$$
  
The series (1) is said to be convergent if for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081300/r0813002.png" /> the series
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The series \eqref{1} is said to be convergent if for any fixed $n$ the series
  
<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/r/r081/r081300/r0813003.png" /></td> </tr></table>
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$$\sum_{m=1}^\infty u_{mn}=a_n$$
  
 
converges and if also the series
 
converges and if also the series
  
<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/r/r081/r081300/r0813004.png" /></td> </tr></table>
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$$\sum_{n=1}^\infty a_n$$
  
converges. The sum of the latter is also called the sum of the repeated series (1). The sum
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converges. The sum of the latter is also called the sum of the repeated series \eqref{1}. The sum
  
<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/r/r081/r081300/r0813005.png" /></td> </tr></table>
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$$s=\sum_{n=1}^\infty a_n=\sum_{n=1}^\infty\left(\sum_{m=1}^\infty u_{mn}\right)$$
  
of the repeated series (1) is the [[Repeated limit|repeated limit]] of the partial sums
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of the repeated series \eqref{1} is the [[Repeated limit|repeated limit]] of the partial sums
  
<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/r/r081/r081300/r0813006.png" /></td> </tr></table>
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$$s_{mn}=\sum_{k=1}^n\sum_{l=1}^mu_{kl},$$
  
i.e.
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''i.e.''
  
<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/r/r081/r081300/r0813007.png" /></td> </tr></table>
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$$s=\lim_{n\to\infty}\lim_{m\to\infty}s_{mn}.$$
  
 
If the [[Double series|double series]]
 
If the [[Double series|double series]]
  
<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/r/r081/r081300/r0813008.png" /></td> </tr></table>
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$$\sum_{m,n=1}^\infty u_{mn}$$
  
 
converges and the series
 
converges and the series
  
<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/r/r081/r081300/r0813009.png" /></td> </tr></table>
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$$\sum_{m=1}^\infty u_{mn}$$
 
 
converges, then the repeated series (1) converges and it has the same sum as the double series . The condition of this theorem is fulfilled, in particular, if the double series
 
  
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converges, then the repeated series \eqref{1} converges and it has the same sum as the double series . The condition of this theorem is fulfilled, in particular, if the double series
 
converges absolutely.
 
converges absolutely.
 
 
 
====Comments====
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964) (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964) {{ZBL|0124.28302}} (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR>
 +
</table>

Latest revision as of 12:59, 20 March 2023

A series whose terms are also series:

$$\sum_{n=1}^\infty\left(\sum_{m=1}^\infty u_{mn}\right).\label{1}\tag{1}$$

The series \eqref{1} is said to be convergent if for any fixed $n$ the series

$$\sum_{m=1}^\infty u_{mn}=a_n$$

converges and if also the series

$$\sum_{n=1}^\infty a_n$$

converges. The sum of the latter is also called the sum of the repeated series \eqref{1}. The sum

$$s=\sum_{n=1}^\infty a_n=\sum_{n=1}^\infty\left(\sum_{m=1}^\infty u_{mn}\right)$$

of the repeated series \eqref{1} is the repeated limit of the partial sums

$$s_{mn}=\sum_{k=1}^n\sum_{l=1}^mu_{kl},$$

i.e.

$$s=\lim_{n\to\infty}\lim_{m\to\infty}s_{mn}.$$

If the double series

$$\sum_{m,n=1}^\infty u_{mn}$$

converges and the series

$$\sum_{m=1}^\infty u_{mn}$$

converges, then the repeated series \eqref{1} converges and it has the same sum as the double series . The condition of this theorem is fulfilled, in particular, if the double series converges absolutely.

References

[a1] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) Zbl 0124.28302 (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Repeated series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Repeated_series&oldid=16645
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article