# Infinite series and summation on infinite sets (summation laws III and IV)

The three main laws, all present for finite series and summaton on finite sets, are:

- Linearity (part I and part II)
- Triangle inequality
- Comparison test

In addition, summation on sets has additional laws:

- Substitution (via bijection)
- Set union
- Fubini's theorem

Substitution and set union laws have some, albeit limited, correspondence with index shifting and range merging present only for standard series.

### Moving from finite to infinite

#### Infinite series

For infinite series, the triangle inequality and the comparison test laws both have tweaks from their finite counterparts.

- Triangle inequality becomes the absolute convergence test. The law includes a condition based on the series convergence.
- The comparison test requires the less series to be less absolutely, so as to rule out non-convergence.

#### Infinite sets

The definition of summation on infinite sets only covers absolute
convergence, as conditional convergence cannot be defined due to the ability to
rearrange the summation to achieve arbitary sums. As a result *
we cannot define the triangle inequality law.* The comparison test also
would get a bit weaker--Tao doesn't even list it, so maybe it's not possible
or useful.

TODO: add the actual laws