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

\[ \left| \sum_{n=p}^{q} a_n \right| \le \varepsilon \text{ for all } p,q \ge N \]

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 \( \lim_{n \rightarrow \infty}a_n = 0 \). 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.

The zero test is an if-then relationship. Convergence of the corresponding sequence to zero is required for a series to be convergent, but the convergence of the sequence to zero is not a sufficient criteria.

Example

The sequence \( a_n := 1 \) is convergent, but does not converge to 0, so the series \( \sum_{n=1}^{\infty}a_n \) must be divergent.

The sequence \( a_n := (-1)^n \) is divergent, so the series \( \sum_{n=1}^{\infty} (-1)^n \) is divergent.

The sequence \( a_n := \frac{1}{n} \) converges to zero; however, the series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is divergent.

Source

p167