State without proof the Binomial Series.
The Binomial Series is:
$$(1+x)^r = \sum_{n=0}^\infty \binom{r}{n} x^n$$
where $\binom{r}{n}$ is the binomial coefficient, given by:
$$\binom{r}{n} = \frac{r(r-1)(r-2)\cdots(r-n+1)}{n(n-1)(n-2)\cdots1}$$
for $n\ge1$, and $\binom{r}{0}=1$.
The Binomial Series can be stated as follows:
For any real number x and any positive integer n, the binomial series expansion for (1 + x)^n is given by:
(1 + x)^n = 1 + nx + (n(n-1)/2!) * x^2 + (n(n-1)(n-2)/3!) * x^3 + ... + ((n(n-1)(n-2)...(n-k+1))/k!) * x^k + ...
Where k is any non-negative integer, and n! denotes the factorial of n.