Tannery's theorem

Last updated July 02, 2023

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 $S_{n}=\sum _{k=0}^{\infty }a_{k}(n)$ and suppose that $\lim _{n\to \infty }a_{k}(n)=b_{k}$ . If $|a_{k}(n)|\leq M_{k}$ and $\sum _{k=0}^{\infty }M_{k}<\infty$ , then $\lim _{n\to \infty }S_{n}=\sum _{k=0}^{\infty }b_{k}$ .^[2]^[3]

Proofs

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

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 $e^{x}$ are equivalent. Note that

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

\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}.

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References

↑ Loya, Paul (2018). Amazing and Aesthetic Aspects of Analysis. Springer. ISBN 9781493967957.
↑ 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.
1 2 Hofbauer, Josef (2002). "A Simple Proof of $1+1/2^{2}+1/3^{2}+\cdots ={\frac {\pi ^{2}}{6}}$ and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR 2695334.

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[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.

[:0-3] 1 2 Hofbauer, Josef (2002). "A Simple Proof of $1+1/2^{2}+1/3^{2}+\cdots ={\frac {\pi ^{2}}{6}}$ and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR 2695334.

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