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

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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