What series is this?

Consider the expansion,

$\cos x =1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+.....$

$\cos x -1=-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+.....$

$1- \cos x =\dfrac{x^2}{2!}-\dfrac{x^4}{4!}+\dfrac{x^6}{6!}-.....$

Integrating on both the sides from 0 to 2,

$\displaystyle \int_0 ^2 (1- \cos x)dx =\int_0 ^2 \Big ( \dfrac{x^2}{2!}-\dfrac{x^4}{4!}+\dfrac{x^6}{6!}-..... \Big ) dx$

$2 - \sin 2 = \dfrac{2^3}{3.2!}-\dfrac{2^5}{5.4!}+\dfrac{2^7}{7.6!}-.....$

$\therefore 2- \sin 2 = \dfrac{8}{3!}-\dfrac{32}{5!}+\dfrac{128}{7!}-.....$

Hence, $\color{#3D99F6}{ \boxed{ \alpha =2, \beta=2}}$

Relevant wiki: Maclaurin Series

$\begin{aligned} S & = \frac 8{3!} - \frac {32}{5!} + \frac {128}{7!} - \frac {512}{9!} + \cdots \\ & = \frac {2^3}{3!} - \frac {2^5}{5!} + \frac {2^7}{7!} - \frac {2^9}{9!} + \cdots \\ & = 2 - \color{#3D99F6} \left(\frac {2}{1!} - \frac {2^3}{3!} + \frac {2^5}{5!} - \frac {2^7}{7!} + \cdots \right) & \small \color{#3D99F6} \text{Maclaurin's series} \\ & = 2 - \color{#3D99F6} \sin 2 \end{aligned}$

$\implies \alpha + 19 \beta - 1 = 2+19(2) - 1= \boxed{39}$