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.[1]

Statement

Let and suppose that . If and , then .[2][3]

Proofs

Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space1.

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 are equivalent. Note that

Define . We have that and that , so Tannery's theorem can be applied and

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References

  1. Loya, Paul (2018). Amazing and Aesthetic Aspects of Analysis. Springer. ISBN 9781493967957.
  2. Koelink, edited by Mourad E.H. Ismail, Erik (2005). Theory and applications of special functions a volume dedicated to Mizan Rahman. New York: Springer. p. 448. ISBN 9780387242330.CS1 maint: extra text: authors list (link)
  3. Hofbauer, Josef (2002). "A Simple Proof of 1 + 1/22 + 1/32 + ⋯ = π2/6 and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR 2695334.

Generalizations of Tannery's theorem

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