Math and science::Analysis::Tao::08. Infinite sets

# 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
• [...] test

• Substitution (via bijection)
• Set union
• [...] 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 [...] test. The law includes a condition based on the series convergence.
• The comparison test requires the less series to be less [...], so as to rule out non-convergence.

#### Infinite sets

The definition of summation on infinite sets only covers [...], as [...] 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.