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.{{cite book |last1=Loya |first1=Paul |title=Amazing and Aesthetic Aspects of Analysis |date=2018 |publisher=Springer |isbn=9781493967957 |url=https://books.google.com/books?id=Q45aDwAAQBAJ&q=Tannery's%20theorem&pg=PA216 |language=en}}

Statement

Let $S_n = \sum_{k=0}^\infty a_k(n)$ and suppose that $\lim_{n\to\infty} a_k(n) = b_k$ . If $|a_k(n)| \le M_k$ and $\sum_{k=0}^\infty M_k < \infty$ , then $\lim_{n\to\infty} S_n = \sum_{k=0}^{\infty} b_k$ .{{cite book |last1= |first1= |title=Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman |date=2005 |publisher=Springer |isbn=9780387242330 |editor-last=Ismail |editor-first=Mourad E. H. |location=New York |page=448 |editor-last2=Koelink |editor-first2=Erik}}{{Cite journal|last=Hofbauer|first=Josef|date=2002|title=A Simple Proof of $1 + 1/2^2 + 1/3^2 + \cdots = \frac{\pi^2}{6}$ and Related Identities|journal=The American Mathematical Monthly|volume=109|issue=2|pages=196–200|doi=10.2307/2695334|jstor=2695334}}

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.

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

: $\lim_{n\to\infty} \sum_{k=0}^\infty {n \choose k} \frac{x^k}{n^k}
=\sum_{k=0}^\infty \lim_{n\to\infty} {n \choose k} \frac{x^k}{n^k}
=\sum_{k=0}^\infty \frac{x^k}{k!}
= e^x.$

References

Category:Theorems in mathematical analysis

Category:Limits (mathematics)