# Disassembling series into series of positive and negative terms [duplicate]

Consider the series $$\sum_{n=0}^{\infty} a_{n}$$. Suppose $$A_+ := \{n ∈ N : a_n ≥ 0\}$$ and $$A_- := \{n ∈ N : a_n < 0\}$$. Can I always disassemble the series as follows?

$$\sum_{n=0}^{\infty} a_{n} = \sum_{n \in A_+} a_{n} + \sum_{n \in A_-} a_{n}$$

Can I make implications such as "the series $$\sum_{n \in A_-} a_{n}$$ is convergent if and only if $$\sum_{n=0}^{\infty} a_{n}$$ and $$\sum_{n \in A_+} a_{n}$$ are convergent"? I am a bit confused because the equality seems like a rearrangement, which can be done safely when the series is absolutely convergent. Please help me understand why we can dismantle the series like that (if we can).

• You cannot always disassemble the series like that as the original series may be convergent while the two others are not. Consider for example $a_n=(-1)^n\frac{1}{n}$. Commented Aug 16, 2022 at 7:59
• @jakobdt I see, thanks for your reply. But if any 2 of the 3 series above is convergent, then I can disassemble the series like that and claim that the third one must also be convergent, right?
– Aid
Commented Aug 16, 2022 at 8:26
• Yes. To make it clear you could define $b_n=a_n$ for $n\in A_+$ and $b_n=0$ otherwise, and similarly $c_n=a_n$ for $n\in A_-$ and $c_n=0$ otherwise. Then $a_n=b_n+c_n$, or perhaps more relevant, $\sum_{n=1}^N a_n=\sum_{n=1}^N b_n+\sum_{n=1}^N c_n$. Commented Aug 16, 2022 at 8:29
• @AndrewD.Hwang I read that one, but saw another solution where they disassemble the series into series of positive and negative terms, which confused me
– Aid
Commented Aug 16, 2022 at 14:20