grandes-ecoles 2012 QII.B.3

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Evaluation of a Finite or Infinite Sum

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $A_n = \sum_{k=0}^n a_k$ and $\widetilde{a}_n = \frac{A_n}{n+1}$.
Calculate $\widetilde { v } _ { n }$ (arithmetic mean of the numbers $v _ { 0 } , \ldots , v _ { n }$).

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $A_n = \sum_{k=0}^n a_k$ and $\widetilde{a}_n = \frac{A_n}{n+1}$.

Calculate $\widetilde { v } _ { n }$ (arithmetic mean of the numbers $v _ { 0 } , \ldots , v _ { n }$).

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.D.1 QII.D.2 QII.D.3 QII.E.1 QII.E.2 QII.E.3 QII.E.4 QII.E.5 QII.E.6