Binomial series (radius of convergence)

[Fernando Revilla]

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]

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 $+\infty$.