grandes-ecoles 2017 QI.B.3

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences
We consider the sequence $\left(I_{n}\right)_{n \in \mathbb{N}}$ defined by $I_{0}=0$ and $\forall n \in \mathbb{N}^{*}, I_{n}=\int_{0}^{+\infty} \ln\left(1+\frac{x}{n}\right) \mathrm{e}^{-x} \mathrm{~d} x$.
a) Show that the sequence $(I_{n})$ is well defined and belongs to $E$.
b) Using integration by parts, show that the sequence $(I_{n})$ belongs to $E^{c}$ and give its convergence rate. 
We consider the sequence $\left(I_{n}\right)_{n \in \mathbb{N}}$ defined by $I_{0}=0$ and $\forall n \in \mathbb{N}^{*}, I_{n}=\int_{0}^{+\infty} \ln\left(1+\frac{x}{n}\right) \mathrm{e}^{-x} \mathrm{~d} x$.

a) Show that the sequence $(I_{n})$ is well defined and belongs to $E$.

b) Using integration by parts, show that the sequence $(I_{n})$ belongs to $E^{c}$ and give its convergence rate.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 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.D.1 QII.D.2 QII.D.3 QII.D.4 QII.D.5 QII.D.6