Math Archive 6

$\begin{aligned} J_n & = \int_0^\frac{\pi}{2} x^n \cos x \, dx \quad \quad \small \color{#3D99F6}{\text{By integration by parts}} \\ & = \left[x^{n} \sin x \right]_0^\frac{\pi}{2} + n \int_0^\frac{\pi}{2} x^{n-1} \sin x \, dx \\ & = \left(\frac{\pi}{2} \right)^n + n \left[x^{n-1} \cos x \right]_0^\frac{\pi}{2} - n(n-1) \int_0^\frac{\pi}{2} x^{n-2} \cos x \, dx \\ & = \left(\frac{\pi}{2} \right)^n - n(n-1)J_{n-2} \\ \Rightarrow J_n + n(n-1)J_{n-2} & = \left(\frac{\pi}{2} \right)^n \\ \frac{J_n + n(n-1)J_{n-2}}{n!} & = \frac{\left(\frac{\pi}{2} \right)^n}{n!} \\ \frac{J_n}{n!} + \frac{J_{n-2}}{(n-2)!} & = \frac{\left(\frac{\pi}{2} \right)^n}{n!} \\ \sum_{n=2}^\infty \left(\frac{J_n}{n!} + \frac{J_{n-2}}{(n-2)!} \right) & = \sum_{n=2}^\infty \frac{\left(\frac{\pi}{2} \right)^n}{n!} \\ \Rightarrow S & = \sum_{n=\color{#3D99F6}{0}}^\infty \frac{\left(\frac{\pi}{2} \right)^n}{n!} \color{#3D99F6}{- 1 - \frac{\pi}{2}} \\ & = e^\frac{\pi}{2} - 1 - \frac{\pi}{2} \\ & = \boxed{2.239} \end{aligned}$

Just an hint: First do the sum, then do the integral. It's much easier!