Math Archive 6

Calculus Level 5

Let \(\displaystyle \, J_{n}=\int_{0}^{\pi /2 }x^{n}\cos(x) \, dx\), where \(n\) is a non-negative integer, and \[ S = \sum_{n=2}^{\infty}\left(\dfrac{J_{n}}{n!}+\dfrac{J_{n-2}}{(n-2)!}\right)\] Find the value of \(S\) up to three decimal places.


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