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

Proofs

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

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

${\displaystyle \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}=\lim _{n\to \infty }\sum _{k=0}^{n}{n \choose k}{\frac {x^{k}}{n^{k}}}.}$

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

${\displaystyle \lim _{n\to \infty }\sum _{k=0}^{\infty }{n \choose k}{\frac {x^{k}}{n^{k}}}=\sum _{k=0}^{\infty }\lim _{n\to \infty }{n \choose k}{\frac {x^{k}}{n^{k}}}=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=e^{x}.}$

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 ${\displaystyle 1+1/2^{2}+1/3^{2}+\cdots ={\frac {\pi ^{2}}{6}}}$ and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR   2695334.