Proof: show that this series converges absolutely to an arbitrary real number This should be a consequence of a well-known result from the theory of conditionally convergent series:
Theorem: Suppose $\sum a_n$ is a conditionally convergent series. Then for every $\beta \in \mathbb{R}$ there exists a partition $\mathbb{N}=\bigcup_{i=1}^{\infty}B_i$, where each set $B_i$ is finite, and the series $\sum_i(\sum_{n\in B_i}a_n)$ converges absolutely to $\beta$.
I'm looking for the proof of this theorem. Maybe someone here could post it or give some references to specific books. You're welcome.

If $\sum_{n\geq 0}a_n$ is conditionally convergent but not absolutely convergent, by the Riemann series theorem there is some permutation $\sigma$ of $\mathbb{N}$ such that $\sum_{n\geq 0}a_{\sigma(n)}$ is conditionally convergent to $\beta$. That means:
$$ \lim_{N\to +\infty}\left(\beta-\sum_{n=0}^{N}a_{\sigma(n)}\right)=0. \tag{1}$$
For any $m\in\mathbb{N}_{>0}$, define $N(m)$ as:
$$ N(m) = \max\left\{N : \left|\beta-\sum_{n=0}^{N}a_{\sigma(n)}\right|\geq \frac{1}{2^m}\right\}\tag{2}$$
then define $B_1,B_2,\ldots$ as:
$$ B_1=\{\sigma(0),\ldots,\sigma(N(1))\},\qquad B_2=\{\sigma(N(1)+1),\ldots,\sigma(N(2))\},\qquad\ldots \tag{3}$$
I leave to you to check that this construction actually works.
