Tannery's theorem

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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.[1]

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].[2][3]

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.[3]

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]

References

  1. Loya, Paul (2018) (in en). Amazing and Aesthetic Aspects of Analysis. Springer. ISBN 9781493967957. https://books.google.com/books?id=Q45aDwAAQBAJ&q=Tannery's%20theorem&pg=PA216. 
  2. Ismail, Mourad E. H., ed (2005). Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman. New York: Springer. p. 448. ISBN 9780387242330. 
  3. 3.0 3.1 Hofbauer, Josef (2002). "A Simple Proof of [math]\displaystyle{ 1 + 1/2^2 + 1/3^2 + \cdots = \frac{\pi^2}{6} }[/math] and Related Identities". The American Mathematical Monthly 109 (2): 196–200. doi:10.2307/2695334.