\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)

Zero tail, and the zero test, propositions

A property of a convergent series is a diminishing tail. This is expressed formally as follows.

Let \( \sum_{n=m}^{\infty}a_n \) be a formal series of real numbers. Then \( \sum_{n=m}^{\infty}a_n \) converges if and only if, for every real number \( \varepsilon > 0 \), there exists an integer \( N \ge m \) such that

This proposition, by itself, is a little difficult to work with, as computing the partial sums at the tail might not be easy. However, there are a number of corollaries, the first of which is the zero test.

Zero test

Let \( \sum_{n=p}^{\infty}a_n \) be a convergent series of real numbers. Then we must have [...]. In other words, if \( \lim_{n \rightarrow \infty}a_n \) is non-zero or divergent, then the series \( \sum_{n=m}^{\infty}a_n \) is divergent.