grandes-ecoles 2016 QI.A.9

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Functional Equations and Identities via Series

We consider a real $\lambda$ and the sequence $\left(u_k = \lambda^k\right)_{k \in \mathbb{N}}$. What is the sequence $\left(v_k\right)_{k \in \mathbb{N}}$ defined by formula $$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j \quad \text{(I.1)}$$ Then verify formula $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j \quad \text{(I.2)}$$

We consider a real $\lambda$ and the sequence $\left(u_k = \lambda^k\right)_{k \in \mathbb{N}}$. What is the sequence $\left(v_k\right)_{k \in \mathbb{N}}$ defined by formula
$$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j \quad \text{(I.1)}$$
Then verify formula
$$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j \quad \text{(I.2)}$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.A.6 QI.A.7 QI.A.8 QI.A.9 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.B.5 QI.B.6 QI.B.7 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3