$$ X = \sum_{i=1}^{\infty} \frac{(-1)^{i+1}}{i} = \log 2$$
If we just look at the series formed from the third terms of the alternating harmonic series, we get
$$Y = -\frac{1}{3} + \frac{1}{6} - \frac{1}{9} + \frac{1}{12} +... $$ $$ = -\frac{1}{3} (1 - \frac{1}{2} + \frac{1}{3} - ...) = -\frac{\log 2}{3}$$
So one would think that the sum of the series in question is
$$ X - Y = \frac{4}{3}\log 2 = 0.9241962407465937$$
But numerical evaluation gives me one fourth of that $0.2310485601873915$.
What am I doing wrong?
Thanks