Binomial Theorem

In the discrete case, the Binomial Theorem is

$(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k}\quad\quad\quad(1)$

where ${n \choose k}$ is the binomial coefficient.

which may be proven with induction.

In the case continuous case, where $r$ is any complex number and $| x / y | < 1$ , or $r$ is a positive integer,

${(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^k y^{r-k}\quad\quad\quad(2)}$

Where : ${r \choose k}={1 \over k!}\prod_{n=0}^{k-1}(r-n)=\frac{r(r-1)(r-2)\cdots(r-(k-1))}{k!}\,$

Which may be proven by taking the power series of the left hand expression.

As useful form of this holds for the reciprocal power

$\frac{1}{(1-x)^r}=\sum_{k=0}^\infty {r+k-1 \choose k} x^k \equiv \sum_{k=0}^\infty {r+k-1 \choose r-1} x^k.$

Which the geometric series is a special case of when $r = 1$ .