grandes-ecoles 2024 Q11

grandes-ecoles · France · mines-ponts-maths1__mp Sequences and Series Evaluation of a Finite or Infinite Sum
Recall that $x$ is a fixed element of $]0;1[$. Deduce that:
$$\frac { \pi } { \sin ( \pi x ) } = \frac { 1 } { x } - \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } x } { n ^ { 2 } - x ^ { 2 } }$$
Recall that $x$ is a fixed element of $]0;1[$. Deduce that:

$$\frac { \pi } { \sin ( \pi x ) } = \frac { 1 } { x } - \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } x } { n ^ { 2 } - x ^ { 2 } }$$