Now I know that there are other questions on the radius of convergence for the power series of binomial expansion, but they do not answer my question.
I already know that the binomial exapansion is expressed as $$(1+x)^r = \sum_{n=0}^r\binom{r}{n}x^n$$
And the power series I am talking about is $$ f(x) := \sum_{n=0}^\infty\binom{r}{n}x^n = 1 + rx + {{r(r-1)\over2}x^2} + \dots$$
I think that this can also be written as $ lim_{r\to\infty}(1+x)^r $. The radius of convergence of the power series above is well known as 1. However, if I substitute $x=1/2$ in $ lim_{r\to\infty}(1+x)^r $, the limit does not converge which contradicts the fact that the radius of convergence is 1. Can anyone explain to me what's going on here? Thanks.
You seem to be confused. The generalized binomial expansion states that $$(1+x)^r=\sum_{n=0}^\infty\binom{r}{n}x^n$$ for $|x|\lt1$ where $r\in\mathbb{R}$. This also requires the definition $$\binom{r}{n}=\frac{r(r-1)\cdots(r-n+1)}{n!}$$ With this definition we have that $\binom{r}{n}$ is zero when $n\gt r$ and $r\in\mathbb{N}_0$ which causes the summation to simplify to $$(1+x)^r=\sum_{n=0}^r\binom{r}{n}x^n$$ when $r\in\mathbb{N}_0$.