Given the power series $$\sum_{n=0}^\infty\frac{(-1)^n}{2^n\sqrt{n}}(x-1)^n,$$ I found using the ratio test that the radius of convergence is $R=1/(1/2)=2$, but I'm a little confused as to why this infinite sum even exists since the first term, for $n=0$, involves dividing by zero. So from a naive point of view, it would seem that the infinite sum should not converge. Does it actually converge?
Mathematica tells me $R=2$ also, but it also complains about division by zero if I try to sum to infinity. This could be a limit of Mathematica, however.
It could be the sum does converge even with the division by zero since the sum is infinite, and as a Taylor series represents some perfectly finite analytic function. Furthermore, by analogy, I think there are certain integrands which diverge at $a$, say, yet whose definite integrals converge over $[a,b]$, say, although I can't remember an explicit example right now.