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?

You may not rearrange the series if it's not absolutely convergent. Otherwise you can get results like these, and like $\sum_{n=0}^\infty (-1)^k = \frac{1}{2}$, which both are false (the last one being the famous Grandi's series, https://en.wikipedia.org/wiki/Grandi%27s_series). All of this is in the Riemann series theorem (https://en.wikipedia.org/wiki/Riemann_series_theorem). Basically, that's the reason why your result is incorrect, and I hope this helps you!
