**Corollary (of [[Every rearrangement of a series with positive terms has the same limit|rearrangement lemma]]):** If $\sum a_{n}$ is [[Absolutely Convergent Series|absolutely convergent]] then every rearrangement has the same limit. Note that $(a_{n})$ is convergent but we can't apply rearrangement lemma directly since we do not have that $a_{n} \geq 0$. **Proof** (similar to [[Absolutely Convergent Series is Convergent]]): Define $\begin{gather} b_{n} =\begin{cases} a_{n} & a_{n} > 0, \\ 0 & a_{n} \leq 0; \end{cases} &c_{n}= \begin{cases} 0 & a_{n} \geq 0, \\ -a_{n} & a_{n} <0. \end{cases}\end{gather}$ Note that $b_{n} \leq |a_{n}|$, $c_{n} \leq |a_{n}|,$ and $a_{n} = b_{n} -c_{n}.$ By rearrangement lemma, we have $\sum_{n=1}^{\infty} b_{\sigma(n)} = \sum_{n=1}^{\infty}b_{n} \quad \text{and} \quad \sum_{n=1}^{\infty} c_{\sigma(n)} = \sum_{n=1}^{\infty}c_{n}$ Since $a_{\sigma(n)} = b_{\sigma(n)} - c_{\sigma(n)}$ it follows that $\sum a_{\sigma(n)} = \sum b_{\sigma(n)} - \sum c_{\sigma(n)} = \sum b_{n} - \sum c_{n} = \sum a_{n}$