Sequences and Series/Multiple limits

Theorem (interchanging summation and integration):

Let be a measure space, and let be a sequence of functions from to , where or . If either of the two expressions


converges, so does the other, and we have


Proof: Regarding the summation as integration over with σ-algebra and counting measure, this theorem is an immediate consequence of Fubini's theorem, given that integration and summation are defined pointwise.

Theorem (interchanging summation and real differentiation):

Let be a sequence of continuously differentiable functions from an open subset of to . Suppose that both


converge for all , and that for all there exists and a sequence in such that

and .


for all .