grandes-ecoles 2011 QIV.A.1

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

Let $g$ be a piecewise continuous increasing function on $[ 0,1 ]$.
By noting $\int _ { 0 } ^ { 1 } = \int _ { 0 } ^ { 1 / 2 } + \int _ { 1 / 2 } ^ { 1 }$, show that

if $n \equiv 1 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \geqslant 0$;
if $n \equiv 3 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \leqslant 0$.

Let $g$ be a piecewise continuous increasing function on $[ 0,1 ]$.

By noting $\int _ { 0 } ^ { 1 } = \int _ { 0 } ^ { 1 / 2 } + \int _ { 1 / 2 } ^ { 1 }$, show that
\begin{itemize}
  \item if $n \equiv 1 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \geqslant 0$;
  \item if $n \equiv 3 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \leqslant 0$.
\end{itemize}

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2