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

1 Answer
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:

#(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#.