Does this power series really converge?

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.

• Is it not even a well defined series. The question of convergence arises only if the first term is omitted. Mar 4, 2019 at 11:40
• indeed, this power series is not defined as the $0$-th term is not defined, in particular you need to either get rid of it or force it to be special term. however, I assume you just want to kill that one off. Mar 4, 2019 at 11:41
• Convergence only concerns the tail of the series, not any particular term. Your series should probably start with $n=1$ since it isn’t well-defined otherwise. The individual terms must certainly exist before you can even begin to consider questions of convergence.
– MPW
Mar 4, 2019 at 11:42
• @KaviRamaMurthy So as it stands, including the $n=0$ term, to ask what its radius of convergence $R$ is would be meaningless?
– gone
Mar 4, 2019 at 11:48
• @Antinous Yes. The question does not make sense. Mar 4, 2019 at 11:50

I think that $$\sum_{n=0}^\infty\frac{(-1)^n}{2^n\sqrt{n}}(x-1)^n$$ is a typo and it should read $$\sum_{n=1}^\infty\frac{(-1)^n}{2^n\sqrt{n}}(x-1)^n$$.
The radius of convergence of $$\sum_{n=1}^\infty\frac{(-1)^n}{2^n\sqrt{n}}(x-1)^n$$ is indeed $$=2.$$