Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. Let $g$, $g^+$, $g^-$, $P$, $Q$ be as defined in II.E.1--II.E.3. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$. Establish the existence of an integer $N _ { 1 } > 0$ such that for every integer $N \geqslant N _ { 1 }$, $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } P \left( x _ { N } ^ { n } \right) \geqslant \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t - \varepsilon$$ and $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } Q \left( x _ { N } ^ { n } \right) \leqslant \int _ { 0 } ^ { 1 } Q ( t ) \mathrm { d } t + \varepsilon$$
Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. Let $g$, $g^+$, $g^-$, $P$, $Q$ be as defined in II.E.1--II.E.3. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$.
Establish the existence of an integer $N _ { 1 } > 0$ such that for every integer $N \geqslant N _ { 1 }$,
$$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } P \left( x _ { N } ^ { n } \right) \geqslant \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t - \varepsilon$$
and
$$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } Q \left( x _ { N } ^ { n } \right) \leqslant \int _ { 0 } ^ { 1 } Q ( t ) \mathrm { d } t + \varepsilon$$