I am trying to prove that if a power series $f(z) = \sum_{n=1}^\infty c_n z^n$ has radius of convergence $R = 1$, and $c_n > 0$ for all $n \in \mathbb{N}$, then $z=1$ is a singular point of $f$.
I was told that I should "use binomial theorem and expand $z^n$ around $1/2$." But what exactly does that mean?
My guess was to write: $$f(z+\frac12) = \sum_{n=1}^\infty c_n (z+\frac12)^n$$ And then apply binomial theorem to $(z+\frac12)^n$ and check the behavior at $z = 1/2$ to show that it is not convergent there (at least I assume that would be the approach for showing it is singular)