Find the sum of the following expression...

$\begin{aligned} S & = \frac 1{2!-1!} - \frac 1{4!-3!} + \frac 1{6!-5!} - \frac 1{8!-7!} + \cdots \\ & = \frac 1{1!(2-1)} - \frac 1{3!(4-1)} + \frac 1{5!(6-1)} - \frac 1{7!(8-1)} + \cdots \\ & = \frac 1{1\cdot 1!} - \frac 1{3 \cdot 3!} + \frac 1{5\cdot 5!} - \frac 1{7\cdot 7!} + \cdots \end{aligned}$

Introducing $x^n$ , where $n$ is non-negative integers, as follows:

$\begin{aligned} S(x) & = \frac x{1\cdot 1!} - \frac {x^3}{3 \cdot 3!} + \frac {x^5}{5\cdot 5!} - \frac {x^7}{7\cdot 7!} + \cdots \\ & = \int \left( \frac 1{1!} - \frac {x^2}{3!} + \frac {x^4}{5!} - \frac {x^6}{7!} + \cdots \right) dx \\ & = \int \frac {\sin x}x dx \end{aligned}$

$\displaystyle \implies S = \int_0^1 \frac {\sin x}x dx = \operatorname{Si} (1) \approx \boxed{0.946}$ , where $\operatorname{Si}(\cdot)$ denotes the sine integral.