Let $T=\{z\in \mathbb{C}:|z|=1\}$ and $D=\{z\in \mathbb{C}:|z|<1\}$. Let $R=$ $$\{z_0\in T: \text{there exists a holomorphic function}\ f\ \text{in open neighborhood}\ U_{z_0} \ \text{of}\ z_0\ \text{such that}\ f(z)=\sum_{n=0}^{\infty}z^{4n}\}.$$ Then which of the following is true?
(1) $T\subset R$
(2) $R$ contains infinitely many points of $T$.
I have no idea how to start this problem. Kindly help.
$$\dfrac 1{1-z^4}=\sum_{n\ge0}z^{4n} $$ is holomorphic at each point of $T$ except $\pm1,\pm i$, the roots of $1-z^4$.
This clear by checking its derivative. Thus $2)$ is true.
Whenever a function is holomorphic at a point, it's holomorphic on a disk around that point.