Cauchy's limit theorem

If the arithmetic means in Cauchy's limit theorem are replaced by weighted arithmetic means those converge as well. More precisely for sequence ${\displaystyle (a_{n})}$ with ${\displaystyle a_{n}\to a}$ and a sequence of positive real numbers ${\displaystyle (p_{n})}$ with ${\displaystyle {\frac {1}{p_{1}+\cdots +p_{n}}}\to 0}$ one has ${\displaystyle {\frac {p_{1}a_{1}+\cdots +p_{n}a_{n}}{p_{1}+\cdots +p_{n}}}\to a}$.[2][1]

This result can be used to derive the Stolz–Cesàro theorem, a more general result of which Cauchy's limit theorem is a special case.[2]

For the geometric means of a sequence a similar result exists. That is for a sequence ${\displaystyle (a_{n})}$ with ${\displaystyle a_{n}>0}$ and ${\displaystyle a_{n}\to a}$ one has ${\displaystyle {\sqrt[{n}]{a_{1}\cdot a_{2}\cdot \cdots \cdot a_{n}}}\ \to a}$.[2][1]

The arithmetic means in Cauchy's limit theorem are also called Cesàro means. While Cauchy's limit theorem implies that for a convergent series its Cesàro means converge as well, the converse is not true. That is the Cesàro means may converge while the original sequence does not. Applying the latter fact on the partial sums of a series allows for assigning real values to certain divergent series and leads to the concept of Cesàro summation and summable series. In this context Cauchy's limit theorem can be generalised into the Silverman–Toeplitz theorem.[1][4]

Let ${\displaystyle \varepsilon >0}$ and ${\displaystyle N\in \mathbb {N} }$ such that ${\displaystyle |a_{k}-a|\leq {\tfrac {\varepsilon }{2}}}$ for all ${\displaystyle k\geq N}$. Due to ${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{N}(a_{k}-a)=0}$ there exists a ${\displaystyle M\in \mathbb {N} }$ with ${\displaystyle \left|{\frac {1}{n}}\sum _{k=1}^{N}(a_{k}-a)\right|\leq {\frac {\varepsilon }{2}}}$ for all ${\displaystyle n\geq M}$ .

Now for all ${\displaystyle n\geq \max(N,M)}$ the above yields:

${\displaystyle {\begin{aligned}\left|{\frac {1}{n}}\left(\sum _{k=1}^{n}a_{k}\right)-a\right|&=\left|{\frac {1}{n}}\sum _{k=1}^{n}(a_{k}-a)\right|=\left|{\frac {1}{n}}\sum _{k=1}^{N}(a_{k}-a)+{\frac {1}{n}}\sum _{k=N+1}^{n}(a_{k}-a)\right|\\&\leq \left|{\frac {1}{n}}\sum _{k=1}^{N}(a_{k}-a)\right|+{\frac {1}{n}}\sum _{k=N+1}^{n}|a_{k}-a|\leq {\frac {\varepsilon }{2}}+{\frac {(n-N)\varepsilon }{2n}}\leq \varepsilon .\end{aligned}}}$[2]

• Sen-Ming: Note on Cauchy's Limit Theorem. In: The American Mathematical Monthly, Vol. 57, No. 1 (Jan., 1950),pp. 28–31 (JSTOR)

• Cesàro means and Cauchy's limit theorem at SOS math
• Cesàro Mean - proof of Cauchy's limit theoren at the ProofWiki