Ok but $a_n/a_{n+1} = (n+1)/(n-1/2) = 1 - (3/2)/(n-1/2)$. So $n(a_n/a_{n+1} - 1) = (3/2) n/(n-1/2)$ which is bounded below.
Raabe's test shows you $\sum_n \binom{1/2}{n}$ indeed does converge. Which means the Taylor expansion of $\sqrt{1 - z}$ at $z = 0$ does absolutely converge on $|z| \leq 1$ :)
Bounded below by $3/2 > 1$, I mean -- that's the important part.
@Wave What about at $z = 1$? Is it analytic there?
@TedShifrin: The reason I thought it ought to be true is that $z = 1$ is a branch point singularity, as you said. The coefficients don't grow nearly as badly enough as $1/(1 - z)$, where it has to for the pole at $z = 1$ to occur, which in turn is what destroys convergence at the circle of convergence.
@Wave No, you won't just be able to see it. But you can differentiate/integrate the series and actually solve for the function represented by this series. Then you can tell.
I should have a look someday. I remembered $\sqrt{1 - z}$ because a friend of mine asked me the same question you did in undergrad complex analysis, and another friend gave your example. I didn't like that, so I convinced him $\sqrt{1 - z}$ will work.
I think I'm having trouble finding the proper mathematical allegory for something. I think way I have is closer to a relation than a map, but it's kind of both. I'm going to be abusing notation and concepts here as I figure out what I'm actually talking about, anyone feel free to correct me.
I have a case in which there are a set of tokens $T$ which can be used to buy products $P$, it's a one way process. The tokens themselves can be considered products $T \subseteq P$. Would I be able to say that $p_1 \geq p_2$ is a relation over $P$ describing "$p_1$ can be exchanged for $p_2$"? If so, t…
