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Calculus Level 5

\[ \large \sum_{n=1}^\infty (-1)^n \left( \ln 2 - \dfrac1{n+1} - \dfrac1{n+2} - \cdots - \dfrac1{n+n} \right)^2 \]

The value of the series above is equal to

\[ \large \dfrac Ga + \dfrac{\pi^b } c - \dfrac \pi d \ln f - \dfrac gh (\ln j)^i \; , \]

where \(a,b,c,d,f,g,h,i\) and \(j\) are positive integers with \(\gcd(g,h) = 1\) and \(G\) denotes the Catalan's constant.

Find \(a+b+c+d+f+g+h+i+j\).

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