---
tag: 750MCA763X
kind: concept
name: Strong Cesàro summability
msc: [40F05, 40G05]
sources:
  - url: https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation
---

## Summary

Strong Cesàro summability asks that the partial sums of a series stay close to a value $s$ on average, measured by the mean of $\abs{s_k - s}^q$.
Ordinary [[Cesàro summation|N8ZGG7EW6J]] only asks that the mean of $s_k - s$ tends to $0$, which allows positive and negative deviations to cancel.
For exponent $q \ge 1$, strong Cesàro summability implies Cesàro summability to the same value.
It is used in the theory of Fourier series and is closely tied to [[statistical convergence|ZW6EHSMANH]].

## Explanations

### L1 human

Grandi's series $1 - 1 + 1 - \cdots$ has partial sums $1, 0, 1, 0, \dots$, whose average tends to $1/2$.
But each partial sum is at distance $1/2$ from $1/2$, so the partial sums are never close to the value.
Strong summability forbids this: the average distance from the value must itself tend to $0$.
So the partial sums must be near $s$ most of the time, not only in balance.