grandes-ecoles 2017 QVA

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series

We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$
Using an enclosure by integrals, determine an asymptotic equivalent of $U _ { n } ( p ) = \sum _ { k = 0 } ^ { p } k ^ { n }$, with $n \geqslant 1$ fixed, as $p$ tends to $+ \infty$.

We fix $n \in \mathbb { N }$. We define the linear map:
$$\begin{aligned}
\Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\
P ( X ) & \mapsto P ( X + 1 ) - P ( X )
\end{aligned}$$

Using an enclosure by integrals, determine an asymptotic equivalent of $U _ { n } ( p ) = \sum _ { k = 0 } ^ { p } k ^ { n }$, with $n \geqslant 1$ fixed, as $p$ tends to $+ \infty$.

Paper Questions

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