Does an unconditionally convergent series of complex numbers converge absolutely?

Question

Suppose $\sum a_n$ is a series of complex numbers, if $\sum a_n$ and its every rearrangement all converge to the same sum, does $\sum a_n$ converge absolutely?

Answer 1

Yes, since $\mathbb{C}$ has finite dimension. You may find this question helpful; quoting from the answer:

Theorem. (Dvoretsky-Rogers, 1950) Every unconditionally convergent series in a Banach space $X$ is absolutely convergent if and only if $X$ is finite dimensional.

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To be clear: the above theorem is good to know, but overkill in your case: you want only the "easy" direction of this theorem. Here it goes:

1. Since $\sum_n a_n$ converges unconditionally, then so do $\sum_n \operatorname{Re}(a_n)$ and $\sum_n \operatorname{Im}(a_n)$.

2. These two are real-valued series, so by the result on $\mathbb{R}$ (which is given by (the contrapositive of) Riemann's rearrangement theorem) $\sum_n \operatorname{Re}(a_n)$ and $\sum_n \operatorname{Im}(a_n)$ are both absolutely convergent.

3. Now, by the triangle inequality this immediately implies that $\sum_n a_n$ is absoltuely convergent as well: indeed, $$\sum_n \lvert a_n\rvert \leq \sum_n \lvert \operatorname{Re}(a_n)\rvert+\sum_n \lvert \operatorname{Im}(a_n)\rvert < \infty$$