grandes-ecoles 2012 QII.A.2

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Power Series Expansion and Radius of Convergence

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence. We assume that the associated power series $\sum a _ { n } x ^ { n }$ has radius of convergence $R _ { a } = 1$ and that the sum $f$ of this series satisfies $f ( x ) \sim \frac { 1 } { 1 - x }$ when $x \rightarrow 1, x < 1$.
Deduce an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not converging to 1.

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence. We assume that the associated power series $\sum a _ { n } x ^ { n }$ has radius of convergence $R _ { a } = 1$ and that the sum $f$ of this series satisfies $f ( x ) \sim \frac { 1 } { 1 - x }$ when $x \rightarrow 1, x < 1$.

Deduce an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not converging to 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