# 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

- or

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

- and

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

- and .

Then

for all .

**Proof:**