grandes-ecoles 2011 QIV.C.1

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

Show that, for all integers $n , p \geqslant 1$ $$\left| \frac { a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) } { a _ { 2 p } f ^ { ( 2 p ) } ( n ) } \right| = \frac { ( \alpha + 2 p ) ( \alpha + 2 p - 1 ) S ( 2 p + 2 ) } { 4 n ^ { 2 } \pi ^ { 2 } S ( 2 p ) }$$ where $f$ is defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$.

Show that, for all integers $n , p \geqslant 1$
$$\left| \frac { a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) } { a _ { 2 p } f ^ { ( 2 p ) } ( n ) } \right| = \frac { ( \alpha + 2 p ) ( \alpha + 2 p - 1 ) S ( 2 p + 2 ) } { 4 n ^ { 2 } \pi ^ { 2 } S ( 2 p ) }$$
where $f$ is defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$.

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