Tannery's theorem

Short description: Mathematical analysis 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 [math]\displaystyle{ S_n = \sum_{k=0}^\infty a_k(n) }[/math] and suppose that [math]\displaystyle{ \lim_{n\to\infty} a_k(n) = b_k }[/math]. If [math]\displaystyle{ |a_k(n)| \le M_k }[/math] and [math]\displaystyle{ \sum_{k=0}^\infty M_k \lt \infty }[/math], then [math]\displaystyle{ \lim_{n\to\infty} S_n = \sum_{k=0}^{\infty} b_k }[/math].

Proofs

Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space [math]\displaystyle{ \ell^1 }[/math].

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 [math]\displaystyle{ e^x }[/math] are equivalent. Note that

[math]\displaystyle{ \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}. }[/math]

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

[math]\displaystyle{ \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. }[/math]