What is wrong with the "proof" for $\ln(2) =\frac{1}{2}\ln(2)$? I have got a question which is as follows:

Is  $\ln(2)=\frac{1}{2}\ln(2)$??

The following argument seems suggesting that the answer is yes:

We have the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$
  which has a mathematically determined value $\ln(2)=0.693$. 
Now, let's do some rearrangement:
$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}-\frac{1}{12}......$$
   $$
 (1-\frac{1}{2})-\frac{1}{4}+(\frac{1}{3}-\frac{1}{6})-\frac{1}{8}+(\frac{1}{5}-\frac{1}{10})-\frac{1}{12}.......$$
  $$\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-\frac{1}{12}......$$
   $$\frac{1}{2}(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}-\frac{1}{12}......)$$
$$\frac{1}{2}\ln(2).$$


I know that mathematics can't be wrong, and I have done something wrong. But here is my question: where does the argument above go wrong?

In your new series
$$\left(\frac 11 - \frac 12\right) - \frac 14 + 
\left(\frac 13 - \frac 16\right) - \frac 18 + 
\left(\frac 15 - \frac 1{10}\right) - \frac 1{12} +
\left(\frac 17 - \frac 1{14}\right) - \frac 1{16} \dots$$
Terms of the form $\frac{1}{2z - 1}$ occur 1/3 of the time instead of 1/2 the time like they do in the original series, so they are underweighted.
Terms of the form $\frac{1}{2z}$ occur 2/3 of the time instead of 1/2 the time like they do in the original series, so they are overweighted.
By changing the density at which those terms appear, you are effectively converting the sum from
$$\lim_{n \to \infty} \sum_{k = 1}^n \frac{1}{2k - 1} - \sum_{k = 1}^n \frac{1}{2k}$$
into 
$$\lim_{n \to \infty} \sum_{k = 1}^{n \cdot 2/3} \frac{1}{2k - 1} - \sum_{k = 1}^{n \cdot 4/3} \frac{1}{2k}$$
which naturally gives a smaller value since the subtracted terms have increased density.  If instead you rearrange the series without changing the density of the subseries, then you'll get an equal result.
