---
tag: N6Z14G81VF
kind: concept
name: Linear and stable summation methods
msc: [40C99, 40D05]
specializes: [8X25Q4XYMN]
sources:
  - url: https://en.wikipedia.org/wiki/Divergent_series
---

## Summary

A [[summation method|8X25Q4XYMN]] is linear when it respects sums and scalar multiples of series.
It is stable when deleting or adding a first term changes the sum by that term.
These two properties, together with [[regularity|29C7C41A52]], are the ones usually expected of a reasonable method.
They are strong enough to force values: any linear and stable method that sums $1 - 1 + 1 - \cdots$ must give it the value $\frac12$.

## Explanations

### L1 human

Ordinary sums obey two familiar rules: the sum of two series is the sum of their sums, and $a_0 + a_1 + a_2 + \cdots$ equals $a_0$ plus the sum of the rest.
Linear and stable methods keep these rules for the new, generalized sums.
For $x = 1 - 1 + 1 - \cdots$ these rules give $x = 1 - x$, so $x = \frac12$.
For $1 + 1 + 1 + \cdots$ they would give $x = 1 + x$, which is impossible, so no such method sums this series to a finite number.