Show that for any real number c, it is possible to rearrange the terms of the series $\sum_{n=1}^\infty (-1)^{n+1} (1/n)$ Show that for any real number c, it is possible to rearrange the terms of the series $\sum_{n=1}^\infty (-1)^{n+1} (1/n)$ so that the sum is exactly c.  

Hints:
Suppose $c>0$. Add the positive terms of the series, one by one, i.e. do the addition $1+1/3+1/5+...$ and stop till this rises above $c$.
Then, start adding the negative terms i.e. $-1/2-1/4-1/6-...$ to the above so that the result just drops below $c$.
Repeat this step with the remaining positive and negative terms.
