The harmonic alternative Two

Calculus Level 4

\[\left( \frac11 + \frac12 + \frac13 \right) - \left(\frac14 + \frac15 + \frac16 \right) + \left(\frac 17 + \frac18 + \frac19 \right) - \left(\frac1{10} + \frac1{11} + \frac1{12} \right) + \ldots\]

Let \(S\) denote the infinite sum above, evaluate \(S\) to 3 decimal places.

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