grandes-ecoles 2013 QII.B.4

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Evaluation of a Finite or Infinite Sum

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Show that, for all $\theta \in ] 0 , \pi [$, $$\sum _ { k = 1 } ^ { + \infty } \frac { \sin ( k \theta ) } { k } = \frac { \pi - \theta } { 2 }$$

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.

Show that, for all $\theta \in ] 0 , \pi [$,
$$\sum _ { k = 1 } ^ { + \infty } \frac { \sin ( k \theta ) } { k } = \frac { \pi - \theta } { 2 }$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.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 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.B.6 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.D.1 QIII.D.2 QIII.D.3 QIII.D.4 QIII.D.5 QIII.D.6 QIII.D.7 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B QIV.C.1 QIV.C.2