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Every partial sum in the "rearranged" series has missing terms. If these terms sum to a non-zero value then the value of the series has been changed. You CAN do this with the Riemann Zeta function because it is absolutely convergent (for real arguments greater than 1) and the sum of the discarded series is zero.

http://lesliegreen.byethost3.com/articles/rearrange.pdf

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