grandes-ecoles 2011 QI.A.3

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

For every real $\alpha > 1$, show that $1 \leqslant S ( \alpha ) \leqslant 1 + \frac { 1 } { \alpha - 1 }$, where $S ( \alpha ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { \alpha } }$.

For every real $\alpha > 1$, show that $1 \leqslant S ( \alpha ) \leqslant 1 + \frac { 1 } { \alpha - 1 }$, where $S ( \alpha ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { \alpha } }$.

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