\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

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

In addition, summation on sets has additional laws:

• 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.