Sum of alternating harmonic series

Edit: As @FamousBlueRaincoat pointed out below, this question was based on an error in a wikipedia article.

The Wikipedia article on the harmonic series gives the following "proof without words" that the alternating harmonic series $1-1/2 +1/3 -1/4 + \cdots$ converges to $\log (2)$:

$$\begin{equation} (1/1)(1/1 - 1/2) + (1/2)(2/3 -2/4) + (1/4)(4/5 - 4/6 + 4/7 - 4/8) + \cdots = \log (2). \end{equation}$$

Can anyone explain why the sum on the left is $\log (2)$? I don't see it right now.

Edit: my goal is specifically to understand this "proof without words" that the alternating harmonic series converges to $\log(2)$.