grandes-ecoles 2022 Q21

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

Let $a > 0$, $I = [-a, a]$, and $$f : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{1+x^2} \end{array}.$$ Show that $f$ is of class $\mathcal { C } ^ { \infty }$ and that, for all $k$ in $\mathbb { N }$ and all $t \in \left] - \pi / 2 , \pi / 2 \right[$, $$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

Let $a > 0$, $I = [-a, a]$, and
$$f : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{1+x^2} \end{array}.$$
Show that $f$ is of class $\mathcal { C } ^ { \infty }$ and that, for all $k$ in $\mathbb { N }$ and all $t \in \left] - \pi / 2 , \pi / 2 \right[$,
$$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

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