Math and science::Analysis::Tao::07. Series

# 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

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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.