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Why does this series have a different sum when its terms are rearranged?

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The problem is:

Give an example of a convergent series such that, when the terms are rearranged, the series sums to a different value.

A solution is:

enter image description here

Although everything makes sense in this solution, I don't get how what it claims is possible. How can a series have two different sums?

Is this caused by the fact that subtraction is not commutative or associative? Hence, in fact, we really have two different series here?


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