Converging series

Fri, Apr 21, 2017

Let \(-1 < x < 1\),

\[ 1 + x + x^2 + x^3 + x^4 + \cdots = \frac{1}{1-x}. \]

Proof

\[ \begin{aligned} (1 - x)(1 + x + x^2 + x^3 + x^4 + \cdots + x^k) &= 1 - x^{k+1}, \\ \lim_{k \to \infty} (1 - x)(1 + x + x^2 + x^3 + x^4 + \cdots + x^k) &= \lim_{k \to \infty} 1 - x^{k+1}, \\ (1 - x)(1 + x + x^2 + x^3 + x^4 + \cdots) &= 1. \end{aligned} \]

A fun way to prove it for \(x = \frac1b\) with \(b \in \mathbb{N}_0\) is to note that for \(a = b-1\)

\[ 1.111\ldots_b = \frac{a.aaa\ldots_b}{a} = \frac{10_b}{a} = \frac{b}{a} = \frac{b}{b-1} = \frac{1}{1-x}. \]