grandes-ecoles 2015 QIV.B.4

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula
We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$, and $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$. We have $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$.
Find again the value of $S _ { 2 }$ already calculated in I.F.3. 
We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$, and $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$. We have $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$.

Find again the value of $S _ { 2 }$ already calculated in I.F.3.
Paper Questions
QI.A.1 QI.A.2 QI.B QI.C.1 QI.C.2 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QI.E QI.F.1 QI.F.2 QI.F.3 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.C.4 QII.C.5 QII.C.6 QII.C.7 QII.C.8 QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIII.D.3 QIV.A QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2