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### The generalised binomial theorem

The generalised binomial theorem was discovered by Isaac Newton (1642–1727). It states that, for \(|x|<1\),

\[ (1+x)^r = \sum_{k=1}^{ \infty}\dfrac{r(r-1)\dotsm(r-k+1)}{k!} x^k \]where \(r\) is any real number.

For \(r = -1\), this is

\[ \dfrac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \dotsb. \]The expansion for \((1-x)^{-1}\) is

\[ \dfrac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \dotsb. \]These are examples of infinite geometric series. We know that they are only convergent for \(|x| < 1\).