# Polylog

Calculus Level 5

$\large \sum _{k=1}^{\infty }\left[\text{Li}_{2k}\left(\frac 14 \right) - \frac 14\right]=\frac{A}{B}-\frac{C}{D}\ln 2+\frac{C}{E}\ln 3$

The equation above holds true for positive integers $$A,B,C,D$$ and $$E$$, where $$\gcd(A,B)=\gcd(C,D)=\gcd(C,E)=1$$.

Find $$A+B+C+D+E$$.

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

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