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As I thought, this is text book material. Whittaker & Watson: A Course of Modern Analysis, 4th ed. Chapter 2 example 6 has the sum for (1/2)log 2 which looks as if it is an exam question from 1908. And section 2.4 cites a reference to Dirichlet in 1837.
At each step in the sequence generation, one positive and two negative terms are created. Every partial sum has twice as many negative as positive terms. The claim is that none are omitted since the series is infinite, and yet at any partial sum step 1/3rd of the elements from the original series have been lost.
Infinity doesn't correct the problem as at every subsequent step the problem gets worse! An infinite number of terms does not mean ALL terms.