Integration of Infinite Series$\int {{{\left( {\frac{1}{{2!}} - \frac{1}{{3!}} + \frac{1}{{4!}} - \frac{1}{{5!}} + .........} \right)}^x}dx} = ?$

We have, ${e^{ - 1}} = 1 + ( - 1) + \frac{{{{( - 1)}^2}}}{{2!}} + \frac{{{{( - 1)}^3}}}{{3!}} + \frac{{{{( - 1)}^4}}}{{4!}} + ..........$

The given integral = $\int {{{({e^{ - 1}})}^x}dx}  = \int {{e^{ - x}}dx}  =  - {e^{ - x}} + C$