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 $|a_{k}(n)|\leq M_{k}$ 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

$\lim _{n\to \infty }\sum _{k=0}^{\infty }{n \choose k}{\frac {x^{k}}{n^{k}}}=\sum _{k=0}^{\infty }\lim _{n\to \infty }{n \choose k}{\frac {x^{k}}{n^{k}}}=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=e^{x}.$