View source for Repeated series

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A [[Series|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|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|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====
<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>