Binomial Theorem

Tags

math

Refers to the expansion of some expression to the power \( n \).

\begin{align} (a+b)^n &= a^n + {n \choose 1} a^{n-1} b + {n \choose 2} a^{n-2} b^2 + \ldots {n \choose k} a^{n-r} b^r + \ldots b^n \\

      & \text{where} \\; {n \choose r} = {}^n C\_r = \frac{n!}{r!(n-r)!}

\end{align}

\begin{align} (1+x)^n = 1 + nx + \frac{n (n-1)}{1 \times 2} x^2 + \ldots + \frac{n (n-1) \ldots (n-r+1)}{1 \times 2 \times 3} x^r + \ldots, \quad |x| < 1 \forall n \in \mathbb{R} \end{align}