Tannery's theorem explained

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 ^{inf ty} _{k =0} a _{k(n)} and suppose that \lim _{n\toinfty} a _{k(n)} = b _{k} . If |a _{k(n)|} \le M _{k} and \sum ^{inf ty} _{k =0} M _{k} < inf ty , then \lim _{n\toinfty} S _{n} = \sum ^{inf ty} _{k =0} b _{k} .^{[2] [3]}

Proofs

\ell ^{1} .

An elementary proof can also be given.

Example

e ^{x} are equivalent. Note that\lim _{n\toinfty} \left( 1 + (x)/(n) \right) ^{n} = \lim _{n\toinfty} \sum ^{n} _{k =0} {n \choose k} (x ^{k})/(n ^{k}) .

Define

a _{k(n)} = {n \choose k} (x ^{k})/(n ^{k}) . We have that |a _{k(n)|} \leq (|x| ^{k})/(k!) and that \sum ^{inf ty} _{k =0} (|x| ^{k})/(k!) = e ^{|x|} < inf ty , so Tannery's theorem can be applied and\lim _{n\toinfty} \sum ^{inf ty} _{k =0} {n \choose k} (x ^{k})/(n ^{k}) = \sum ^{inf ty} _{k =0} \lim _{n\toinfty} {n \choose k} (x ^{k})/(n ^{k}) = \sum ^{inf ty} _{k =0} (x ^{k})/(k!) = e ^{x.}

Notes and References

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