Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 17 / c / iii / Solution

Start with the supplied Gaussian integral $I(0)=\sqrt{\pi }/2$ and part (i)'s $I(1)=1/2$. Repeated use of the Gaussian moment recurrence givesThe even indices inherit the $\sqrt{\pi }$ factor from $I(0)$, while the odd indices inherit rational values from $I(1)$.