grandes-ecoles 2012 QI.B.2

grandes-ecoles · France · centrale-maths1__pc Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions

Let $n \in \mathbb { N } ^ { * }$ and $x \in [ 0,1 ]$. We consider the sum $$S ( x ) = \sum _ { k = 0 } ^ { n } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } .$$ a) Write the Cauchy-Schwarz inequality in the space $\mathbb { R } ^ { n + 1 }$ equipped with its canonical inner product. b) Using question I.A.4, deduce that $S ( x ) \leqslant \frac { 1 } { 2 \sqrt { n } }$.

Let $n \in \mathbb { N } ^ { * }$ and $x \in [ 0,1 ]$. We consider the sum
$$S ( x ) = \sum _ { k = 0 } ^ { n } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } .$$
a) Write the Cauchy-Schwarz inequality in the space $\mathbb { R } ^ { n + 1 }$ equipped with its canonical inner product.\\
b) Using question I.A.4, deduce that $S ( x ) \leqslant \frac { 1 } { 2 \sqrt { n } }$.

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