Given a non degenerated function $f:\mathbb N^+\to\mathbb Z_2 $. (It doesn't stabilize and become constant for all $n$ larger than some natural number). Define the function $\sigma(i,n,f)=|f^{-1}(i)\cap\mathbb Z_n|\,$ and
$c_n= \left\{ \begin{array}{l} a_{\sigma(o,n,f)}\;\text{ if } f(n)=0\\ b_{\sigma(1,n,f)}\;\text{ if } f(n)=1 \end{array} \right. $
Then $(c_n)_n$ is a shuffle of $(a_n)_n$ and $(b_n)_n$ for each function $f$ as above. All the terms $a_i$ and $b_i$ occur exactly once in the sequence $c_1,c_2,\dots$ in preserved index order.
I would like to formulate this conjecture:
If $\displaystyle a=\sum^\infty_{i=1} a_i\;$ and $\displaystyle \; b=\sum^\infty_{i=1} b_i\;$ are convergent and $c_i$ arise from a shuffle of $a_i$ and $b_i$, then $\displaystyle a+b=\sum^\infty_{i=1} c_i$.
I'm asking for a proof of the conjecture or a counterexample and will award the three first correct answers with a bounty worth 200.
Forall $n$, $\sum_{i=1}^n c_i = \sum_{i=1}^{\sigma(0,n,f)} a_i + \sum_{i=1}^{\sigma(1,n,f)} b_i$.
Since $\sum_{i=1}^\sigma a_i$ converges to $a$ as $\sigma \to \infty$ and $\sigma(0,n,f)$ diverges to infinity as $n \to \infty$, $\sum_{i=1}^{\sigma(0,n,f)} a_i$ converges to $a$ as $n \to \infty$.
Similarly, $\sum_{i=1}^{\sigma(1,n,f)} b_i$ converges to $b$, and so $\sum_{i=1}^n c_i$ converges to $a+b$ as $n \to \infty$.