Let there be a sequence of real numbers $(a_n)_{n\in\mathbb{N}}$, and build out of it a series $(\sum_{j=1}^na_j)_{n\in\mathbb{N}}$ which converges.
Now define a new sequence $(b_n)_{n\in\mathbb{N}}$ via $b_n:=a_{2n-1}+a_{2n}$ for all $n\in\mathbb{N}$.
Is it in general true that $\lim_{n\to\infty}\sum_{j=1}^na_j=\lim_{n\to\infty}\sum_{j=1}^nb_j$ ?
I would say yes, but I am not sure if perhaps absolute convergence of $(\sum_{j=1}^na_j)_{n\in\mathbb{N}}$ is necessary to conclude that. If you do need absolute convergence, where exactly would you use it? It cannot be a re-arrangement argument, because we're not re-arranging the order of the sequence $(a_n)_{n\in\mathbb{N}}$, we are merely regrouping.
The sequence of partial sums of the $b_j$ is a subsequence of the sequence of partial sums of the $a_j$.
Since the sequence of partial sums of the $a_j$ converges, so does the sequence of partial sums of the $b_j$, and to the same value.
Remark: The assumption that the sequence of partial sums of the $a_j$ converges is a very powerful condition. Absolute convergence is not needed.