grandes-ecoles 2020 Q14

grandes-ecoles · France · x-ens-maths-b__mp_cpge Taylor series Derive series via differentiation or integration of a known series

Show that for all $a \in \mathbb{R}$, $$\int_0^a \sin(x^2) \mathrm{d}x = \sum_{n=0}^{+\infty} (-1)^n \frac{a^{4n+3}}{(2n+1)!(4n+3)}$$

Show that for all $a \in \mathbb{R}$,
$$\int_0^a \sin(x^2) \mathrm{d}x = \sum_{n=0}^{+\infty} (-1)^n \frac{a^{4n+3}}{(2n+1)!(4n+3)}$$