How to make an infinite sum equal any number you want
In short: In standard arithmetic, changing the order of numbers doesn't change the total. But with certain infinite series, rearranging the terms can completely rewrite the outcome, allowing you to reach any value imaginable.
The phenomenon relies on conditionally convergent series—sequences where the sum settles on a specific value only because positive and negative terms cancel each other out. If you take the absolute value of these terms, the series would instead grow toward infinity.
The Riemann Rearrangement Theorem proves that these specific series can be reordered to sum to any real number, or even to diverge entirely. For example, while the alternating harmonic series naturally sums to ln(2), a specific rearrangement can force it to equal 1/2 ln(2). In contrast, absolutely convergent series, like the geometric series, remain stable regardless of the order of their terms.
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Sources
- Summary by Intellect · source: The bizarre world of INFINITE rearrangements // Riemann Series Theorem
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