Tannery's theorem

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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 S n = ∑ k = 0 n a k ( n ) {\displaystyle S_{n}=\sum _{k=0}^{n}a_{k}(n)} {\displaystyle S_{n}=\sum _{k=0}^{n}a_{k}(n)} and suppose that lim n → ∞ a k ( n ) = b k {\displaystyle \lim _{n\to \infty }a_{k}(n)=b_{k}} {\displaystyle \lim _{n\to \infty }a_{k}(n)=b_{k}}. If | a k ( n ) | ≤ M k {\displaystyle |a_{k}(n)|\leq M_{k}} {\displaystyle |a_{k}(n)|\leq M_{k}} and ∑ k = 0 ∞ M k < ∞ {\displaystyle \sum _{k=0}^{\infty }M_{k}<\infty } {\displaystyle \sum _{k=0}^{\infty }M_{k}<\infty }, then lim n → ∞ S n = ∑ k = 0 ∞ b k {\displaystyle \lim _{n\to \infty }S_{n}=\sum _{k=0}^{\infty }b_{k}} {\displaystyle \lim _{n\to \infty }S_{n}=\sum _{k=0}^{\infty }b_{k}}.[2][3]

Proofs

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

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 e x {\displaystyle e^{x}} {\displaystyle e^{x}} are equivalent. Note that

lim n → ∞ ( 1 + x n ) n = lim n → ∞ ∑ k = 0 n ( n k ) x k n k . {\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}}}.} {\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}}}.}

Define a k ( n ) = ( n k ) x k n k {\displaystyle a_{k}(n)={n \choose k}{\frac {x^{k}}{n^{k}}}} {\displaystyle a_{k}(n)={n \choose k}{\frac {x^{k}}{n^{k}}}}. We have that | a k ( n ) | ≤ | x | k k ! {\displaystyle |a_{k}(n)|\leq {\frac {|x|^{k}}{k!}}} {\displaystyle |a_{k}(n)|\leq {\frac {|x|^{k}}{k!}}} and that ∑ k = 0 ∞ | x | k k ! = e | x | < ∞ {\displaystyle \sum _{k=0}^{\infty }{\frac {|x|^{k}}{k!}}=e^{|x|}<\infty } {\displaystyle \sum _{k=0}^{\infty }{\frac {|x|^{k}}{k!}}=e^{|x|}<\infty }, so Tannery's theorem can be applied and

lim n → ∞ ∑ k = 0 ∞ ( n k ) x k n k = ∑ k = 0 ∞ lim n → ∞ ( n k ) x k n k = ∑ k = 0 ∞ x k k ! = e x . {\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}.} {\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}.}

References

  1. Loya, Paul (2018). Amazing and Aesthetic Aspects of Analysis. Springer. ISBN 9781493967957.
  2. Ismail, Mourad E. H.; Koelink, Erik, eds. (2005). Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman. New York: Springer. p. 448. ISBN 9780387242330.
  3. Hofbauer, Josef (2002). "A Simple Proof of 1 + 1 / 2 2 + 1 / 3 2 + ⋯ = π 2 6 {\displaystyle 1+1/2^{2}+1/3^{2}+\cdots ={\frac {\pi ^{2}}{6}}} {\displaystyle 1+1/2^{2}+1/3^{2}+\cdots ={\frac {\pi ^{2}}{6}}} and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR 2695334.