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
and suppose that
. If
and
, then
.[2][3]
Proofs
Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space
.
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
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df77bdfdb29402f1ba35d0bb6b25a6bb74685657)
Define
. We have that
and that
, so Tannery's theorem can be applied and
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb8d5256bf9e8b1c482d8764371e3bd1a04556a)
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.
- ^ a b Hofbauer, Josef (2002). "A Simple Proof of
and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR 2695334.