Remarks derived from geometric series

Some Useful Results

• \((1-x)^{-1} = 1 + x + x^2 + \dots + x^r + \dots\) for \(|x| < 1\).
• \((1+x)^{-1} = 1 - x + x^2 - \dots + (-1)^r x^r + \dots\) for \(|x| < 1\).

These are derived from the sum to infinity of a GP.

For example,
\[ \frac{1}{1-x} = S_\infty, \]
where \(S_\infty\) is the sum to infinity of the geometric progression, with first term \(1\) and common ratio \(r=x\).