Why is it true that absolute convergence implies unconditional convergence ? This is interesting, because the sequences (corresponding to each rearrangement) of partial sums are different, still they converge to the same value if the series is absolutely convergent. I want to know the intuitive reason.
[Math] absolute convergence implies unconditional convergence
real-analysis
Best Answer
In my opinion, the intuitive argument should run as follows:
A series is absolutely convergent iff and only if the sequence of the partial sums of the absolute values of its terms (which is always monotonic non-decreasing) is bounded above.
Now consider any re-arrangement of this series of absolute values of the terms of the original series.
If we form a re-arrangement of this series (of absolute values) and consider any given term of its sequence of partial sums, we would find that that term would be less than or equal to some term of the (bounded above) sequence of partial sums of the absolute values of the terms of the original (absolutely convergent) series.
Thus the sequence of the partial sums of this re-arrangement, which is again non-decreasing, is also bounded above.
Hence the re-arrangement is also (absolutely) convergent.