grandes-ecoles 2014 QII.C.6

grandes-ecoles · France · centrale-maths1__psi Taylor series Formal power series manipulation (Cauchy product, algebraic identities)

Let $z \in \mathbb{C}$. Consider the function of the real variable $x$ $$G_z : x \mapsto \sum_{p=0}^{+\infty} \left(x^p(2z - x)^p\right)$$ Deduce (from II.C.5) that $G_z$ admits a Taylor expansion to any order at 0. We denote it $$G_z(x) = \sum_{k=0}^{n} a_k x^k + o\left(x^n\right) \quad x \to 0$$ Determine the coefficients $a_k$ for $k \in \mathbb{N}$.

Let $z \in \mathbb{C}$. Consider the function of the real variable $x$
$$G_z : x \mapsto \sum_{p=0}^{+\infty} \left(x^p(2z - x)^p\right)$$
Deduce (from II.C.5) that $G_z$ admits a Taylor expansion to any order at 0. We denote it
$$G_z(x) = \sum_{k=0}^{n} a_k x^k + o\left(x^n\right) \quad x \to 0$$
Determine the coefficients $a_k$ for $k \in \mathbb{N}$.

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