Tannery's theorem

Template:Short description 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 $S_n=\sum _{k=0}^{\mathrm{\infty }}{a}_k(n)$ and suppose that $\underset{n\to \mathrm{\infty }}{\lim }{a}_k(n)=b_k$. If $|a_k(n)|\le M_k$ and $\sum _{k=0}^{\mathrm{\infty }}{M}_k<\mathrm{\infty }$, then $\underset{n\to \mathrm{\infty }}{\lim }{S}_n=\sum _{k=0}^{\mathrm{\infty }}{b}_k$.

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

$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{x}{n})}^n=\underset{n\to \mathrm{\infty }}{\lim }\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\frac{x^k}{n^k}.$

Define $a_k(n)=(\genfrac{}{}{0ex}{}{n}{k})\frac{x^k}{n^k}$. We have that $|a_k(n)|\le \frac{|x{|}^k}{k!}$ and that $\sum _{k=0}^{\mathrm{\infty }}\frac{|x{|}^k}{k!}=e^{|x|}<\mathrm{\infty }$, so Tannery's theorem can be applied and

$\underset{n\to \mathrm{\infty }}{\lim }\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{k})\frac{x^k}{n^k}=\sum _{k=0}^{\mathrm{\infty }}\underset{n\to \mathrm{\infty }}{\lim }(\genfrac{}{}{0ex}{}{n}{k})\frac{x^k}{n^k}=\sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}=e^x.$