Power series infinity at every point of boundary Is there an example of a power series $f(z)=\sum_{k=0}^\infty a_kz^k$ with radius of convergence $0<R<\infty$ so that $\sum_{k=0}^\infty a_kw^k=\infty$ for all $w$ with $|w|=R$ 
Thank you kindly.

No there is not. In fact, there is no example of such power series $\sum_n a_n z^n$ such that $\sum_n a_n w^n = \infty$ for all $w$ in a set of positive measure in $\partial D$, where $D=\{|z|<R\}$. Indeed, suppose there exists such a power series $f$. By Abel's Theorem, we deduce that $f(z)$ has non-tangential boundary values $\infty$ on a set of positive measure in $\partial D$. This means that $1/f$ is a meromorphic function in $D$ with non-tangential boundary values $0$ on a set of positive measure in $\partial D$, and so $1/f$ is identically zero in $D$, by the Luzin-Privalov Theorem. So $f \equiv \infty$ in $D$, a contradiction.
