I have the following problem. Rearrange the terms in the alternating harmonic series such that the sum is $s \in \mathbb{R}$
I thought about proving it with Riemann's rearrangement theorem, but I am unsure if that is what it really asks (and also unsure about how to exactly prove that). Rather some kind of formula maybe or rule to achieve a given $s$?
Thank you for your help.
Notice that the sum over all positive terms of the alternating harmonic series diverges. The same is true for the sum over all negative terms of the series. Therefore you can choose $k$ such that the sum over the first $k$ positive summands of the alternating harmonic series is greater than $s\in\mathbb{R}$ i.e. \begin{align*} q=\sum_{n=0}^k\frac{1}{2n+1}\geq s. \end{align*} Now pick $k'$ such $q$ minus the first $k'$ negative summands of the alternating harmonic series is smaller then $s$ i.e. \begin{align*} p=q-\sum_{n=0}^{k'}\frac{1}{2n}< s. \end{align*} Notice that you can now find $k''$ such that $p$ plus the $k+1$ to the $k+k''$ positive summands of the alternating harmonic series is greater then $s$. You can keep this pattern going for ever and for that order of summands the sum will obviously converge to $s$.