# How do you find the radius of convergence of the binomial power series?

Sep 21, 2014

The radius of convergence of the binomial series is $1$.

Let us look at some details.

The binomial series looks like this:

${\left(1 + x\right)}^{\alpha} = {\sum}_{n = 0}^{\infty} \left(\begin{matrix}\alpha \\ n\end{matrix}\right) {x}^{n}$,
where

((alpha),(n))={alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}/{n!}

By Ratio Test,

${\lim}_{n \to \infty} | \frac{{a}_{n + 1}}{{a}_{n}} | = {\lim}_{n \to \infty} | \frac{\left(\begin{matrix}\alpha \\ n + 1\end{matrix}\right) {x}^{n + 1}}{\left(\begin{matrix}\alpha \\ n\end{matrix}\right) {x}^{n}} |$

=lim_{n to infty}|{{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)(alpha-n)}/{(n+1)!}x^{n+1}}/{{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}/{n!}x^n}|

by cancelling out all common factors,

$= {\lim}_{n \to \infty} | \frac{\alpha - n}{n + 1} x |$

by pulling $| x |$ out of the limit,

$= | x | {\lim}_{n \to \infty} | \frac{\alpha - n}{n + 1} |$

by dividing the numerator and the denominator by $n$,

$= | x | {\lim}_{n \to \infty} | \frac{\frac{\alpha}{n} - 1}{1 + \frac{1}{n}} | = | x | | \frac{0 - 1}{1 + 0} | = | x | < 1$

Hence, the radius of convergence is $1$.