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$\displaystyle\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n }({ H }_{ n }-\ln { (n+2x) } -\gamma ) } =\dfrac { \gamma }{ a } -\ln { \left( \dfrac { (2x-1)!!\sqrt [ b ]{ \pi } }{ { c }^{ x }x! } \right) }$ If the equation above is true for positive integers $a$ , $b$ , $c$ and $x$ , find $a+b+c$ .

Notations :

• $H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n$ is the $n^\text{th}$ harmonic number .
• $\gamma \approx 0.5772$ is the Euler-Mascheroni constant .
• $!!$ denotes the double factorials function.