### 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 $$|x| < 1$$.

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