grandes-ecoles 2013 QII.C.1

grandes-ecoles · France · centrale-maths1__psi Sequences and Series Functional Equations and Identities via Series

Using Euler's formula, justify that for $( n , k )$ in $\mathbb { N } \times \mathbb { N }$,
$$I _ { n , k } = \sum _ { m = 0 } ^ { k } \frac { A _ { m , k } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { i t ( 2 m - k - n ) } \mathrm { d } t$$
with $A _ { m , k }$ constants to be determined.

Using Euler's formula, justify that for $( n , k )$ in $\mathbb { N } \times \mathbb { N }$,

$$I _ { n , k } = \sum _ { m = 0 } ^ { k } \frac { A _ { m , k } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { i t ( 2 m - k - n ) } \mathrm { d } t$$

with $A _ { m , k }$ constants to be determined.

Paper Questions

QI.A.1 QI.A.2 QI.B QI.C QI.D QII.A QII.B QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.D QII.E.1 QII.E.2 QII.E.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 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2