grandes-ecoles 2022 Q10

grandes-ecoles · France · mines-ponts-maths1__psi Sequences and Series Evaluation of a Finite or Infinite Sum

Using a series expansion under the integral, show that $$\int_{0}^{+\infty} \ln(1-e^{-u}) \mathrm{d}u = -\frac{\pi^2}{6}$$

Using a series expansion under the integral, show that
$$\int_{0}^{+\infty} \ln(1-e^{-u}) \mathrm{d}u = -\frac{\pi^2}{6}$$