The binomial theorem specifies how we can expand expressions of the form \((x+y)^n\).
In the case that \(n\) is a positive integer, it takes the form \[\begin{align*} (x+y)^n &= \sum_{i=0}^n \binom{n}{i} x^i y^{n-i} \\ &= x^n + nx^{n-1} y + \binom{n}{2} x^{n-2} y^2+ \dotsb + y^n. \end{align*}\]In the general case, where \(n\) is not necessarily a positive integer, we have \[ (1+x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \dotsb \] whenever \(\big|x\big|<1\).
