grandes-ecoles 2019 Q9

grandes-ecoles · France · centrale-maths2__pc Taylor series Prove smoothness or power series expandability of a function

Let $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$ and $g$ the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$. Both satisfy $2h^{\prime}(x) = h(x)^2 + 1$. By considering the functions $\arctan f$ and $\arctan g$, show $$\forall x \in I, \quad f(x) = g(x).$$

Let $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$ and $g$ the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$. Both satisfy $2h^{\prime}(x) = h(x)^2 + 1$. By considering the functions $\arctan f$ and $\arctan g$, show
$$\forall x \in I, \quad f(x) = g(x).$$

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