We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $\widetilde{a}_n = \frac{1}{n+1}\sum_{k=0}^n a_k$.
Construct using $\psi$ an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not satisfying property II.3 ($\lim_{n\to\infty} \widetilde{a}_n = 1$).