Conditionally convergent series are convergent, but not absolute convergent. Example: 1-1/2+1/3-1/4+...=ln(2), but 1+1/2+1/3+1/4+...=∞. Riemann proved that the terms of such a series can be rearranged so that the new series converges to an arbitrary number, or even diverges.
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Yes: the commutativity of finite sums doesn't imply it for infinite ones. This is like each term in a convergent sequence having a property, but the limit not having it.
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Actually you can rearrange the series to converge to any real number (and infinities ) or diverge. Intuitively: fix M and start adding positive numbers until the partial sum is greater than M, then add negative numbers until the sum is smaller than M. Repeat.
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i'm not either, but i like to learn also i'm not sure why they went with a weird counterexample; if i understand correctly there are easier ones (https://math.stackexchange.com/questions/2151423/how-come-rearrangement-of-a-convergent-series-may-not-converge-to-the-same-value …)
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ie. conditional convergence
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In machine learning you can still get the same recognition values. Like a rearranged image.
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