grandes-ecoles 2014 QI.B.4

grandes-ecoles · France · centrale-maths1__psi Sequences and series, recurrence and convergence Proof by induction on sequence properties

We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Show that, for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, we have $$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$ One may proceed by induction.

We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$.\\
Show that, for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, we have
$$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$
One may proceed by induction.

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