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#.
