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**Calculus**Level 5

\[\displaystyle\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n }({ H }_{ n }-\ln { (n+2x) } -\gamma ) } =\dfrac { \gamma }{ a } -\ln { \left( \dfrac { (2x-1)!!\sqrt [ b ]{ \pi } }{ { c }^{ x }x! } \right) } \] 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.