grandes-ecoles 2012 QII.E.6

grandes-ecoles · France · centrale-maths1__pc Sequences and series, recurrence and convergence Coefficient and growth rate estimation

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $\widetilde{a}_n = \frac{A_n}{n+1}$ where $A_n = \sum_{k=0}^n a_k$.
Conclude (i.e., prove property II.3: $\lim_{n \to \infty} \widetilde{a}_n = 1$).

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $\widetilde{a}_n = \frac{A_n}{n+1}$ where $A_n = \sum_{k=0}^n a_k$.

Conclude (i.e., prove property II.3: $\lim_{n \to \infty} \widetilde{a}_n = 1$).

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