Cesàro mean

Definition

Let $(a_n)_{n \ge 1}$ be a real sequence. For each integer $n \ge 1$, the $n$-th Cesàro mean of $(a_n)$ is the arithmetic mean of its first $n$ terms, $$\sigma_n = \frac{a_1 + a_2 + \cdots + a_n}{n}.$$

Remarks

The sequence $(-1)^{n+1}$ diverges, but its Cesàro means are $1, 0, 1/3, 0, 1/5, \dots$, and they tend to $0$. If $(a_n)$ converges, the Cesàro means converge to the same limit. Cesàro summation applies the Cesàro means to the partial sums of a series. A series is Cesàro summable with sum $A$ when the means of its partial sums tend to $A$.

Explanations

L1 human

The Cesàro mean is the running average of the sequence. Averaging smooths out oscillations. The sequence $1, 0, 1, 0, \dots$ has no limit, but its running averages tend to $1/2$.