Two to Tango

Algebra Level 2

What is the value of \(S\) if

\[S = \frac{1}{2} - \frac{1}{6} + \frac{1}{4} - \frac{1}{12} + \frac{1}{8} - \frac{1}{20} + \frac{1}{16} - \frac{1}{30} + \frac{1}{32} - \frac{1}{42} + \cdots \]

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