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 $n \in \mathbb{N}$. We denote $A_N = \sum_{k=0}^N a_k$. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$. Let $N_1$ be as in II.E.4. Deduce from the three previous questions that for every integer $N \geqslant N _ { 1 }$ $$1 - 5 \varepsilon \leqslant \left( 1 - x _ { N } \right) A _ { N } \leqslant 1 + 5 \varepsilon$$
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 $n \in \mathbb{N}$. We denote $A_N = \sum_{k=0}^N a_k$. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$. Let $N_1$ be as in II.E.4.
Deduce from the three previous questions that for every integer $N \geqslant N _ { 1 }$
$$1 - 5 \varepsilon \leqslant \left( 1 - x _ { N } \right) A _ { N } \leqslant 1 + 5 \varepsilon$$