Exhibit a power series $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ with radius of convergence 1 and sum $f$, such that $f ( z )$ converges when $z \rightarrow 1 , | z | < 1$ and such that the series $\sum _ { n \geqslant 0 } a _ { n }$ does not converge.