grandes-ecoles 2013 QI.C

grandes-ecoles · France · centrale-maths1__psi Sequences and Series Functional Equations and Identities via Series

Express $G _ { x } ( t + \pi )$ and deduce the following equalities for $n$ in $\mathbb { Z }$ :
$$\varphi _ { n } ( - x ) = ( - 1 ) ^ { n } \varphi _ { n } ( x ) = \varphi _ { - n } ( x )$$
What can be said about the parity of $\varphi _ { n }$ for $n \in \mathbb { Z }$ ?

Express $G _ { x } ( t + \pi )$ and deduce the following equalities for $n$ in $\mathbb { Z }$ :

$$\varphi _ { n } ( - x ) = ( - 1 ) ^ { n } \varphi _ { n } ( x ) = \varphi _ { - n } ( x )$$

What can be said about the parity of $\varphi _ { n }$ for $n \in \mathbb { Z }$ ?

Paper Questions

QI.A.1 QI.A.2 QI.B QI.C QI.D QII.A QII.B QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.D QII.E.1 QII.E.2 QII.E.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2