What if we rearrange the terms of a known series?

Calculus Level 5

\[\large \frac{1}{2}+\frac{1}{4}-\frac{1}{1}+\frac{1}{6}+\frac{1}{8}-\frac{1}{3}+\frac{1}{10}+\frac{1}{12}-\frac{1}{5}+\cdots\]

The series above is obtained by rearranging the terms of the alternating harmonic series. Determine the approximate value of the sum of the series rounded to three decimal places if it converges, otherwise, enter 0.

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