2. a. Show that the radius of convergence of the power series $\sum \frac{E_n}{n!} x^n$ is $\geq 1$. b. For $|x| < 1$, we denote by $f(x)$ the sum of the preceding power series. Prove that $$2f'(x) = f^2(x) + 1, \quad \forall x \in ]-1, 1[$$ c. Deduce that $f(x) = \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{1}{\cos x} + \tan x, \quad \forall x \in ]-1, 1[$, then that $$\frac{1}{\cos x} = \sum_{n=0}^{\infty} \frac{E_{2n}}{(2n)!} x^{2n}, \quad \tan x = \sum_{n=0}^{\infty} \frac{E_{2n+1}}{(2n+1)!} x^{2n+1}, \quad \forall x \in ]-1, 1[$$
2. a. Show that the radius of convergence of the power series $\sum \frac{E_n}{n!} x^n$ is $\geq 1$.\\
b. For $|x| < 1$, we denote by $f(x)$ the sum of the preceding power series. Prove that
$$2f'(x) = f^2(x) + 1, \quad \forall x \in ]-1, 1[$$
c. Deduce that $f(x) = \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{1}{\cos x} + \tan x, \quad \forall x \in ]-1, 1[$, then that
$$\frac{1}{\cos x} = \sum_{n=0}^{\infty} \frac{E_{2n}}{(2n)!} x^{2n}, \quad \tan x = \sum_{n=0}^{\infty} \frac{E_{2n+1}}{(2n+1)!} x^{2n+1}, \quad \forall x \in ]-1, 1[$$