a) For every $z \in \mathbb { C }$ such that $| z | < 1$, justify that the series $\sum _ { p \in \mathbb { N } } a _ { p } z ^ { p }$ is convergent. We denote by $\varphi ( z )$ its sum: $\varphi ( z ) = \sum _ { p = 0 } ^ { \infty } a _ { p } z ^ { p }$. b) For $z \in \mathbb { C }$ such that $| z | < 1$, calculate the product $\left( e ^ { z } - 1 \right) \varphi ( z )$. Deduce that for every $z \in \mathbb { C } ^ { * }$ satisfying $| z | < 1$, we have $\varphi ( z ) = \frac { z } { e ^ { z } - 1 }$. c) Show that $a _ { 2 k + 1 } = 0$ for every integer $k \geqslant 1$. Calculate $a _ { 4 }$.
a) For every $z \in \mathbb { C }$ such that $| z | < 1$, justify that the series $\sum _ { p \in \mathbb { N } } a _ { p } z ^ { p }$ is convergent.
We denote by $\varphi ( z )$ its sum: $\varphi ( z ) = \sum _ { p = 0 } ^ { \infty } a _ { p } z ^ { p }$.
b) For $z \in \mathbb { C }$ such that $| z | < 1$, calculate the product $\left( e ^ { z } - 1 \right) \varphi ( z )$. Deduce that for every $z \in \mathbb { C } ^ { * }$ satisfying $| z | < 1$, we have $\varphi ( z ) = \frac { z } { e ^ { z } - 1 }$.
c) Show that $a _ { 2 k + 1 } = 0$ for every integer $k \geqslant 1$. Calculate $a _ { 4 }$.