Repeated series

A series whose terms are also series:

(1)

The series (1) is said to be convergent if for any fixed the series

converges and if also the series

converges. The sum of the latter is also called the sum of the repeated series (1). The sum

of the repeated series (1) is the repeated limit of the partial sums

i.e.

If the double series

converges and the series

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

converges absolutely.

References

[a1]	K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)