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.
∑
∞
a
( n )
and suppose that
lim
n → ∞
If
k
and
∑
lim
[2][3] Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space
ℓ
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
, so Tannery's theorem can be applied and