Question on a Previous Post about the Sum of Series The post in question is here. In the comments, it is shown further that it converges for $|z|\neq 1$ However, no one answers how to show it converges to an analytic function in this region. Thank you for taking a look.
 A: Given an open set $\Omega\subseteq \Bbb C$, recall that the following conditions are equivalent:


*

*$f:\Omega\to\Bbb C$ is holomorphic on $\Omega$;

*$f:\Omega\to\Bbb C$ is continuous, and for all $z_0\in \Omega$ there is an open disk $\Delta(z_0,r)\subseteq \Omega$ such that $\int_{\gamma} f(z)\,dz=0$ for all (say) piecewise linear closed paths $\gamma:[a,b]\to \Delta(z_0,r)$;

*$f:\Omega\to\Bbb C$ is continuous, and $\int_\gamma f(z)\,dz$ for all $z_0\in\Omega$, all open disks $\Delta(x_0,r)\subseteq \Omega$ and all piecewise linear closed paths $\gamma:[a,b]\to\Delta(z_0,r)$.
This is essentially Morera's theorem plus the fact that holomorphy is a local property. A consequence of this is that if $f_n$ is a sequence of holomorphic functions $\Omega\to \Bbb C$ which converges uniformly on compact subsets of $\Omega$, then their limit $f$ is holomorphic on $\Omega$. In fact let $z_0\in \Omega$, let $K\subseteq\Omega$ be compact and $r>0$ such that $\Delta(z_0,r)\subseteq K^\circ$, and let $\gamma:[a,b]\to\Delta(z_0,r)$ be a piecewise linear closed path. $f$ is obviously continuous. By (3), $\int_\gamma f_n(z)\,dz=0$ for all $n$, so that $$0\le\left\lvert\int_\gamma f(z)\,dz\right\rvert=\left\lvert\int_\gamma f(z)-f_n(z)\,dz\right\rvert\le \int_a^b \lvert f(\gamma(t))-f_n(\gamma(t))\rvert M_\gamma\,dt\le\\\le M_\gamma(b-a)\sup_{z\in K}\lvert f(z)-f_n(z)\rvert\stackrel{n\to\infty}\longrightarrow 0$$
where $M_\gamma$ is a constant that depends only on $\gamma$. Therefore, $\int_\gamma f(z)\,dz=0$ and $f$ satifies (2). Therefore, $f$ is holomorphic.
This applies to the sequence of partial sums, with the open set $\Omega=\{z\in\Bbb C\,:\, \lvert z\rvert\ne 1\}$. For uniform convergence on compact subsets of $\Omega$, you can use Weierstrass M-test and the estimates given in the answers (e.g. user Martín-Blas Pérez Pinilla's)
A: It has a Taylor series that converges to it inside $|z|\lt1$ and a Laurent series that converges to it outside the circle.
