It's kinda weird summation

Calculus Level 5

$\large \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1} \sin(k)}{k^3}$

If the above sum can be represented as $\dfrac{\pi^a - b}{c}$ for positive integers $a$ , $b$ and $c$ , find the value of $a+b+c$ .

Clarification: Angles are measured in radians.