Tannery's theorem

In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery.

Statement

Let and suppose that . If and , then .

Proofs

Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space .

An elementary proof can also be given.

Example

Tannery's theorem can be used to prove that the binomial limit and the infinite series characterizations of the exponential are equivalent. Note that

Define . We have that and that , so Tannery's theorem can be applied and

References