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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


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.