State without proof the Binomial Series.

Answer 1

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 $\binom{r}{0}=1$.