grandes-ecoles 2014 QII.C.5

grandes-ecoles · France · centrale-maths1__psi Sequences and series, recurrence and convergence Series convergence and power series analysis

Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$, and $V_n(z) = U_{n+1}(z,-1)$. Show that there exists a non-empty open disk $\Delta$ with center $O$ included in $\Omega_z$ such that $$\forall Z \in \Delta, \quad \frac{1}{1 - 2zZ + Z^2} = \sum_{n=0}^{+\infty} V_n(z) Z^n = \sum_{p=0}^{+\infty} \left(Z^p(2z - Z)^p\right)$$

Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$, and $V_n(z) = U_{n+1}(z,-1)$.\\
Show that there exists a non-empty open disk $\Delta$ with center $O$ included in $\Omega_z$ such that
$$\forall Z \in \Delta, \quad \frac{1}{1 - 2zZ + Z^2} = \sum_{n=0}^{+\infty} V_n(z) Z^n = \sum_{p=0}^{+\infty} \left(Z^p(2z - Z)^p\right)$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.A.6 QI.A.7 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6