grandes-ecoles 2018 Q23

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Evaluation of a Finite or Infinite Sum

Show that $$\forall k \in \mathbb{N}^{*}, \quad \zeta(k+1) = \frac{1}{k!} \int_{0}^{+\infty} \frac{u^{k}}{\mathrm{e}^{u} - 1} \mathrm{~d}u$$

Show that
$$\forall k \in \mathbb{N}^{*}, \quad \zeta(k+1) = \frac{1}{k!} \int_{0}^{+\infty} \frac{u^{k}}{\mathrm{e}^{u} - 1} \mathrm{~d}u$$