Zero tail, and the zero test, propositions

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

Let $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ be a formal series of real numbers. Then $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ converges if and only if, for every real number $\epsilon >0$, 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 $\sum _{n=p}^{\mathrm{\infty }}{a}_n$ be a convergent series of real numbers. Then we must have [...]. In other words, if $\underset{n\to \mathrm{\infty }}{lim}{a}_n$ is non-zero or divergent, then the series $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ is divergent.