Let $\alpha_n = f^{(n)}(0)$ where $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$. Let $R$ be the radius of convergence of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ and $g$ its sum. Using Taylor's formula with integral remainder, show
$$\forall N \in \mathbb{N}, \forall x \in \left[0, \pi/2\left[, \quad \sum_{n=0}^{N} \frac{\alpha_n}{n!} x^n \leqslant f(x)\right.\right.$$