Sticky Summation (1)

Multiply both sides by $\frac{\sqrt {3}}{2}$ , and the series can be expressed as

$\begin{aligned} \frac{\sqrt {3}}{2} \cdot S & = \sum_{x=0}^\infty \frac {\sin \frac {\pi x}3}{x!} & \small \color{#3D99F6} \text{Using Euler's formula: } e^{i \theta} = \cos \theta +i \sin \theta \\ & = \sum_{x=0}^\infty \frac {\Im \left(e^{i\frac {\pi x}3}\right)}{x!} & \small \color{#3D99F6} \Im (z) \text{ is the imaginary part of complex number }z. \\ & = \Im \left(\sum_{x=0}^\infty \frac {e^{i\frac {\pi}3x}}{x!}\right) \\ & = \Im \left(\exp \left(e^{i\frac {\pi}3}\right) \right) & \small \color{#3D99F6} \exp (z) = e^z \\ & = \Im \left(\exp \left(\cos \frac \pi 3 + i \sin \frac \pi 3 \right) \right) \\ & = \Im \left(\exp \left( \frac 12 + i \frac {\sqrt 3}2 \right) \right) \\ & = \Im \left(e^\frac 12 \cdot e^{i\frac {\sqrt 3}2} \right) \\ & = \Im \left(\sqrt e \left(\cos \frac {\sqrt 3}2 + i\sin \frac {\sqrt 3}2\right) \right) \\ & = \sqrt e \sin \frac {\sqrt 3}2 \end{aligned}$

$\implies X =\frac{3}{4} = \boxed{0.75}$ .