Théorème de Cesaro

Soit $u$ une suite complexe convergeant vers un complexe $l$ . Montrer que
$\frac{1}{n}\sum_{k=0}^{n-1}u_{k}\underset{n\to+\infty}{\longrightarrow}l.$
Soit $f$ une fonction complexe continue telle que $f(x)\underset{x\to+\infty}{\longrightarrow}l$ avec $l\in\mathbb{C}$ . Montrer que
$\frac{1}{x}\int_{0}^{x}f(t)\,\mathrm{d}t\underset{x\to+\infty}{\longrightarrow}l.$
A-t-on la réciproque d’une des propositions précédentes ? Autrement dit, est-ce que la convergence en valeur moyenne implique toujours la convergence de la suite / fonction continue considérée ?

I solved itMark it done Add to my listKeep it in your list

I like this problem1 like

Liked by

Anduril@anduril

Solutions

Reveal solutionsAre you sure? Give it a try first.

Potential issue reported

Solution by AndurilEN

Discussions0 useful votes

Let $(a_n)_{n \in \mathbb{N}^*}$ be a sequence of complex numbers such that $\lim_{n \to \infty} a_n = \ell \in \mathbb{C}$ . Then:

$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n a_k = \ell$

Setting $b_n = a_n - \ell$ , we can assume without loss of generality that $\ell = 0$ .

Let $\varepsilon > 0$ . Since $b_n \to 0$ , there exists $N \in \mathbb{N}^*$ such that:

$\forall k > N, \quad |b_k| \le \frac{\varepsilon}{2}$

For any $n > N$ , split the sum into two parts:

$\left| \frac{1}{n} \sum_{k=1}^n b_k \right| \le \frac{1}{n} \sum_{k=1}^N |b_k| + \frac{1}{n} \sum_{k=N+1}^n |b_k| \le \frac{M_N}{n} + \frac{n - N}{n} \frac{\varepsilon}{2} < \frac{M_N}{n} + \frac{\varepsilon}{2}$

where $M_N = \sum_{k=1}^N |b_k|$ is a fixed constant independent of $n$ .

Since $\lim_{n \to \infty} \frac{M_N}{n} = 0$ , there exists $N' \ge N$ such that:

$\forall n > N', \quad \frac{M_N}{n} < \frac{\varepsilon}{2}$

Thus, for all $n > N'$ :

$\left| \frac{1}{n} \sum_{k=1}^n a_k - \ell \right| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon$

For the second question, same reasoning.

As counterexamples, you can consider $u_{n}=(-1)^{n}$ and $f=sin$ .

Still, there are some converse results such as when the sequence / function is real and monotonous (or more generally if we already know that it has a limit - finite or not).

Report