Real Series

Abstract

In Chap. 4, we have defined the decimal representation for a real number $r \in \mathbb {R}$ as the formal infinite sum:

$$\displaystyle r \sim a_0+\frac {a_1}{10}+\frac {a_2}{10^2}+\frac {a_3}{10^3}+ \ldots =\sum _{j=0}^\infty \frac {a_j}{10^j} \ \text{ or } \ a_0.a_1a_2a_3\ldots , $$

where $a_0 \in \mathbb {Z}$ and $a_j \in \{0,1,2\ldots ,9\}$ for all $j \in \mathbb {N}$ with no recurring 9s in the representation. This sum, which is an infinite sum, was stated formally (hence why the symbol $\sim$ was used).

The petty cares, the minute anxieties, the infinite littles which go to make up the sum of human experience, like the invisible granules of powder, give the last and highest polish to a character.

— William Matthews, poet