Consider the power series $\sum a_n z^n$.Given that $a_n$ converges to $0$, prove that $f(z)$ cannot have pole on the unit circle, where $f(z)$ is the function represented by the power series in the question.
I have thought an answer for it. Since $a_n$ converges to $0$, we can write $\lvert a_n \rvert <1$ for all $n >N_0$. From here, we can say radius of convergence of the power series is bigger than or equal to $1$. If the radius of convergence is bigger than $1$, the series converges on the unit circle. If it is equal to $1$, then points on the unit circle cannot be an isolated singularity. But I am not sure of my answer.