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I am looking for an infinite convergent sum that still converges after reordering the sum but with a different limit. Are there any examples? Thanks.

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    Have you heard of the [Rearrangement Theorem](http://en.wikipedia.org/wiki/Riemann_series_theorem)?2012-12-12

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Any conditionally convergent series (i.e. convergent but not absolutely convergent) will do. A classical example is $ \sum_{n=1}^\infty\frac{(-1)^n}{n}. $