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 $$ S_n = \sum_{k=0}^\infty a_k(n) $$ and suppose that $$ \lim_{n\to\infty} a_k(n) = b_k $$. If $$ |a_k(n)| \le M_k $$ and $$ \sum_{k=0}^\infty M_k < \infty $$, then  $$ \lim_{n\to\infty} S_n = \sum_{k=0}^{\infty} b_k $$.

Proofs
Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space $$\ell^1$$.

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 $$ e^x $$ are equivalent. Note that


 * $$ \lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^n = \lim_{n\to\infty} \sum_{k=0}^n {n \choose k} \frac{x^k}{n^k}. $$

Define $$ a_k(n) = {n \choose k} \frac{x^k}{n^k} $$. We have that $$ |a_k(n)| \leq \frac{|x|^k}{k!} $$ and that $$ \sum_{k=0}^\infty \frac{|x|^k}{k!} = e^{|x|} < \infty $$, so Tannery's theorem can be applied and


 * $$ \lim_{n\to\infty} \sum_{k=0}^\infty {n \choose k} \frac{x^k}{n^k}

=\sum_{k=0}^\infty \lim_{n\to\infty} {n \choose k} \frac{x^k}{n^k} =\sum_{k=0}^\infty \frac{x^k}{k!} = e^x. $$