# On isolated singularity on circle of convergence of a Taylor series [closed]

Suppose that $f(z)$ is analytic in the unit disk $\Delta:\,|z|<1$. Then $f(z)$ has a Taylor series $\sum=\sum a_nz^n$ in the unit disk. Assume that $\sum$ has $R=1$ as its radius of convergence.

Question 1: Suppose that $f(z)$ has isolated singularity at $z=1$ and the coefficients of Taylor series $\sum$ satisfies $a_n\to A$ where $A\neq0$. Is $z=1$ a pole of $f(z)$?

Question 2: Suppose that $f(z)$ has isolated singularity at $z=1$ and the coefficients of Taylor series $\sum$ satisfies $a_n\to \infty$. Is $z=1$ a pole of $f(z)$?

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What have you tried? It's always a good idea to show exactly where you are stuck. –  icurays1 Jan 24 '13 at 23:20
What do you mean by pole when the function is not defined on a punctured neighbourhood of the point? –  mrf Jan 25 '13 at 6:34