How's this related to trig?

Calculus Level 5

\[ \dfrac{7}{5} = x + \dfrac{x^2}{2!} - \dfrac{x^3}{3!} - \dfrac{x^4}{4!} + \dfrac{x^5 }{5!} + \dfrac{x^6}{6!} - \dfrac{x^7}{7!} - \dfrac{x^8}{8!} + ... \]

If the value of \( x \) that satisfies the above equation can be expressed in the form \( 2 \arctan \left( \dfrac{-a + \sqrt{b}}{c}\right) + 2\pi k\) for integer \(k\), with where \( a\) and \(c \) are positive coprime integers and \( b\) is square-free, find \( a + b + c \).

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