grandes-ecoles 2023 Q5

grandes-ecoles · France · polytechnique-maths__fui Taylor series Prove smoothness or power series expandability of a function

Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by
$$\left\{ \begin{array}{l} \varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\ \varphi_{0}(0) = 0 \end{array} \right.$$
a. Show that for all $n \in \mathbb{N}$ there exists a polynomial $P_{n}$ such that for $x \neq 0$ we have
$$\varphi_{0}^{(n)}(x) = P_{n}\left(\frac{1}{x}\right) e^{-1/x^{2}}$$
b. Show that $\varphi_{0}$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by

$$\left\{ \begin{array}{l} 
\varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\
\varphi_{0}(0) = 0
\end{array} \right.$$

a. Show that for all $n \in \mathbb{N}$ there exists a polynomial $P_{n}$ such that for $x \neq 0$ we have

$$\varphi_{0}^{(n)}(x) = P_{n}\left(\frac{1}{x}\right) e^{-1/x^{2}}$$

b. Show that $\varphi_{0}$ is of class $C^{\infty}$ on $\mathbb{R}$.

Paper Questions

QExercise-1 QExercise-2 QExercise-3 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20