grandes-ecoles 2021 Q31

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Power Series Expansion and Radius of Convergence

Using the results of questions 28--30, show that there exists a unique complex sequence $(b_n)_{n \in \mathbb{N}}$ and a real number $r > 0$ such that, for all $z \in \mathbb{C}$, $$0 < |z| < r \Rightarrow \frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n.$$

Using the results of questions 28--30, show that there exists a unique complex sequence $(b_n)_{n \in \mathbb{N}}$ and a real number $r > 0$ such that, for all $z \in \mathbb{C}$,
$$0 < |z| < r \Rightarrow \frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n.$$

Paper Questions

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