It's kinda weird summation

Calculus Level 5

k=1(1)k+1sin(k)k3\large \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1} \sin(k)}{k^3}

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

Clarification: Angles are measured in radians.

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