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.[1]
Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space .
An elementary proof can also be given.[3]
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