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 ℓ1.
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
