grandes-ecoles 2011 QI.B.1

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series

For every real $\alpha$ strictly greater than 1 and for every non-zero natural integer $n$, we set $R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } }$. Using the inequality from question I.A.1, show that $R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + O \left( \frac { 1 } { n ^ { \alpha } } \right)$.

For every real $\alpha$ strictly greater than 1 and for every non-zero natural integer $n$, we set $R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } }$. Using the inequality from question I.A.1, show that $R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + O \left( \frac { 1 } { n ^ { \alpha } } \right)$.

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