factorials and the roots of unity

$\begin{aligned} S & = \frac 1{1!} + \frac {-1}{2!} + \frac 0{3!} + \frac 1{4!} + \frac {-1}{5!} + \frac 0{6!} + \frac 1{7!} + \frac {-1}{8!} + \frac 0{9!} + \cdots \\ & = \frac 2{\sqrt 3} \left(\frac {\sin \frac {2\pi}3}{1!} + \frac {\sin \frac {4\pi}3}{2!} + \frac {\sin 2\pi}{3!} + \frac {\sin \frac {8\pi}3}{4!} + \frac {\sin \frac {10\pi}3}{5!} + \cdots \right) & \small \color{#3D99F6} \text{By Euler's formula: }e^{\theta i} = \cos \theta + i \sin \theta \\ & = \Im \left(\frac 2{\sqrt 3}\left(\frac {e^{\frac {2\pi}3i}}{1!} + \frac {e^{\frac {4\pi}3i}}{2!} + \frac {e^{\frac {6\pi}3i}}{3!} + \frac {e^{\frac {8\pi}3i}}{4!} + \frac {e^{\frac {10\pi}3i}}{5!} + \cdots \right) \right) & \small \color{#3D99F6} \Im (\cdot) \text{ denotes the imaginary part function.} \\ & = \frac 2{\sqrt 3} \Im \left(e^{e^{\frac {2\pi}3i}} - 1\right) \\ & = \frac 2{\sqrt 3} \Im \left(e^{\cos \frac {2\pi}3 + i \sin \frac {2\pi}3 } - 1\right) \\ & = \frac 2{\sqrt 3} \Im \left(e^{-\frac 12 + i \frac {\sqrt 3}2} - 1\right) \\ & = \frac 2{\sqrt 3} \Im \left(\frac 1{\sqrt e}\left(\cos \frac {\sqrt 3}2 + i\sin \frac {\sqrt 3}2 \right) - 1\right) \\ & = \frac 2{\sqrt {3e}} \sin \frac {\sqrt 3}2 \end{aligned}$

Therefore, $A+B+C = 3+3+2 = \boxed 8$ .

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Reference: Euler's formula