The radius of convergence of the binomial series is 1.
Let us look at some details.
The binomial series looks like this:
(1+x)^alpha=sum_{n=0}^infty((alpha),(n))x^n,
where
((alpha),(n))={alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}/{n!}
By Ratio Test,
lim_{n to infty}|{a_{n+1}}/{a_n}|=lim_{n to infty}|{((alpha),(n+1))x^{n+1}}/{((alpha),(n))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}|{alpha-n}/{n+1}x|
by pulling |x| out of the limit,
=|x|lim_{n to infty}|{alpha-n}/{n+1}|
by dividing the numerator and the denominator by n,
=|x|lim_{n to infty}|{alpha/n-1}/{1+1/n}|=|x||{0-1}/{1+0}|=|x|<1
Hence, the radius of convergence is 1.
