What if we rearrange the terms of a known series?

Calculus Level 5

12+1411+16+1813+110+11215+ \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 series rounded to three decimal places if it converges; otherwise, enter 0.

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