grandes-ecoles 2010 QIII.B.1

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Show that, for all $x \in [-1,1]$, the series $$\sum_{n \geqslant 0} \alpha_n F_n(x)$$ is convergent.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.

Show that, for all $x \in [-1,1]$, the series
$$\sum_{n \geqslant 0} \alpha_n F_n(x)$$
is convergent.

Paper Questions

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