# Binomial series (radius of convergence)

#### Fernando Revilla

##### Well-known member
MHB Math Helper
I quote a question from Yahoo! Answers

Is the radius of convergence for all binomial series exactly 1?
I have given a link to the topic there so the OP can see my response.

#### Fernando Revilla

##### Well-known member
MHB Math Helper
If $|x|<1$, the binomial series $\displaystyle\sum_{k=0}^{\infty} \; {\alpha \choose k} \; x^k$ converges absolutely to $(1+x)^{\alpha}$ for any $\alpha\in\mathbb{R}$, but not always the radius of convergence is $1$. For example, if $\alpha$ is a non-negative integer, then the series is finite and the radius of convergence is $+\infty$.