Cesàro Mean
Contents
Theorem
Let $\left\langle{a_n}\right\rangle$ be a sequence of complex numbers.
Suppose that $\left\langle{a_n}\right\rangle$ converges to $\ell$ in $\C$:
- $\displaystyle \lim_{n \to \infty} a_n = \ell$
Then also:
- $\displaystyle \lim_{n \to \infty} \frac{a_1 + \dots + a_n} n = \ell$
Proof
For every fixed integer $n_0$, we write:
- $\displaystyle \left\vert{\frac{a_1 + \dots + a_n} n - \ell}\right\vert \le \frac{\left\vert{a_1 - \ell}\right\vert + \dots + \left\vert{a_n - \ell}\right\vert} {n} \le \frac {n_0 \sup_{k \le n_0} \left\vert{a_k - \ell}\right\vert} n + \sup_{n_0 < k \le n} \left\vert{a_k - \ell}\right\vert$
As $n$ tends to $+\infty$, we get:
- $\displaystyle \limsup_{n \to \infty} \left\vert{\frac{a_1 + \dots + a_n} n - \ell}\right\vert \le \sup_{k > n_0} \left\vert{a_k - \ell}\right\vert$
As $n_0$ tends to $+\infty$, we finally conclude:
- $\displaystyle \limsup_{n \to \infty} \left\vert{\frac{a_1 + \dots + a_n} n - \ell}\right\vert = 0$
$\blacksquare$
More detailed justification and explanation of all the steps
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Remarks
Need to split off the case for normed vector spaces
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
- The theorem and its proof hold in any normed vector space.
- When working with an arbitrary sequence $\left\langle{a_n}\right\rangle$ of real numbers, the same truncation trick leads to:
- $\displaystyle \liminf_{n \to \infty} a_n \le \liminf_{n\to\infty} \frac{a_1 + \dots + a_n} n \le \limsup_{n \to \infty} \frac{a_1 + \dots + a_n} n \le \limsup_{n \to \infty} a_n$
As a corollary, the conclusion of the theorem holds in the real case when $\ell = \pm \infty$.
Source of Name
This entry was named for Ernesto Cesàro.
