Complex analysis power series / derivative

Question

I need some help with this exercise: Show that there is no such power series $f(z)=\sum_{n=0}^{\infty}C_nz^n$ such that:

• $f(z)=1$ for $z=\frac{1}{2}, \frac{1}{3}, \frac{1}{4},\ldots,$
• $f'(0)>0$.

I've been given the following corollary:

If a power series equals zero at all the points of a set with an accumulation point at the origin, then the power series is identically zero.

Answer 1

Since $f(z)-1=0$ for each $z\in\left\{\frac12,\frac13,\frac14,\ldots\right\}$, $f(z)-1$ is the null function. In other words, $f(z)=1$. But then $f'(0)=0$.

Answer 2

The function is equal to $1$ for all values of form $\frac{1}{n}$ ,also the accumulation point of this sequence is zero which is in its domain.This implies that function $f(x)=1$ (by identity theorem) and we know that derivative of constant function is zero so your argument $f'(0)>0$ contraadicts .So there exists no such function.