I hope you can help me on a problem I'm working on a long time.
We had to rearrange the alternating harmonic series
$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$$
in such a way, that we get 5.
I split the series up in two new ones. One with the positive numbers $\Bigl(\sum\frac{1}{2i-1}\Bigl)$ and one with the negative numbers $\Bigl(\sum-\frac{1}{2t}\Bigl)$. I also wrote it down in a short formula but I don't know how the determine $\alpha$:
$$\sum_{n=1}^{\infty}\Biggl(\sum_{i=(n-1)\alpha+1}^{(n-1)\alpha+\alpha} \frac{1}{2i-1} +\sum_{t=n}^{n} -\frac{1}{2t}\Biggl)=5$$
What is the easiest way to determine $\alpha$?
(My idea was to add positive numbers until I'm over 5, then subtract a negative number so we are below 5, add positive until I'm above 5 again, subtract a negative and so on.