Suppose $\sum a_n$ is convergent,can we impose some condition on this series to make the series absolutely convergent?
Except some trivial conditions that it has finitely many positive or negative terms etc.
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1 Answer
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In $\mathbb{R}$ (resp. $\mathbb{C}$) the absolute convergence is equivalent to the condition that for any permutation $\pi \colon \mathbb{N} \rightarrow \mathbb{N}$ the series $\sum_{n=1}^\infty a_{\pi(n)}$ is convergent (known as Riemann rearrangement theorem). In general, you cannot say more! (Of course, you could use e.g. the integral comparison test, if $|a_n|$ can be extenend to an monotically decreasing function $f$ in order to get absolute convergence. However, this is - of course - only a very special case.)
Note that by the Theorem of Dvoretzky-Rogers this statement is false in infinite dimensional banach spaces. (There is an unconditional convergent series which is not absolute convergent.)
