grandes-ecoles 2017 QVE

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Properties and Manipulation of Power Series or Formal Series

We fix $n \in \mathbb { N }$. For all $r \in \mathbb { N }$, there exists a unique polynomial $P _ { r } ( X )$ such that $$\forall p \in \mathbb { N } , \quad \sum _ { k = 1 } ^ { p } k ^ { 2 r + 1 } = P _ { r } \left( \frac { p ( p + 1 ) } { 2 } \right)$$
V.E.1) Determine the leading term in $P _ { r } ( X )$.
V.E.2) Show that for $r \geqslant 1$, $X ^ { 2 }$ divides $P _ { r } ( X )$.
V.E.3) Explicitly give the polynomials $P _ { 1 } ( X )$ and $P _ { 2 } ( X )$.

We fix $n \in \mathbb { N }$. For all $r \in \mathbb { N }$, there exists a unique polynomial $P _ { r } ( X )$ such that
$$\forall p \in \mathbb { N } , \quad \sum _ { k = 1 } ^ { p } k ^ { 2 r + 1 } = P _ { r } \left( \frac { p ( p + 1 ) } { 2 } \right)$$

V.E.1) Determine the leading term in $P _ { r } ( X )$.

V.E.2) Show that for $r \geqslant 1$, $X ^ { 2 }$ divides $P _ { r } ( X )$.

V.E.3) Explicitly give the polynomials $P _ { 1 } ( X )$ and $P _ { 2 } ( X )$.

Paper Questions

QIA QIB QIC QID QIIA QIIB QIIC QIID QIIE QIIF QIIIA QIIIB QIIIC QIIID QIVA QIVB QIVC QVA QVB QVC QVD QVE