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Calculus Level 4

\[ \large \displaystyle\sum _{ k=1 }^{ \infty }{ \frac { \sin { k } }{ { k }^{ 2 } } =\frac { A }{ B } i({ \text{Li} }_{ 2 }({ e }^{ -Ci } }) -{\text{Li} }_{ 2 }({ e }^{ Di })) \]

The equation above holds true for positive integers \(A,B,C\) and \(D\), with \(A,B\) coprime. Find \(A+B+C+D\).

Notation: \({ \text{Li} }_{ n }(a) \) denotes the polylogarithm function, \({ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \frac { { a }^{ k } }{ { k }^{ n } } }. \)

Clarification:\(i=\sqrt{-1}\)

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