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The commutative law doesn't hold for certain series

I'm sure that you know that it doesn't matter in what order you add up a (finite) collection of numbers. No matter how you rearrange them, you'll end up with the same sum. (If the order mattered, nobody could ever balance a checkbook.) This fact is called the commutative law, but weirdly enough, it's not true for certain infinite sums, called conditionally convergent series. ( tex2html_wrap_inline399 is conditionally convergent if it converges, but tex2html_wrap_inline401 diverges.) In fact, it's been proven that if tex2html_wrap_inline399 converges conditionally, you can pick whatever number you like and rearrange the series to sum to that number. I won't give the general proof, but I will give a specific example.

Let's consider the alternating harmonic series tex2html_wrap_inline405 . It converges by the alternating series test, but the sum of its absolute values is the harmonic series, which diverges. Therefore, the alternating harmonic series is conditionally convergent. Let's call its sum S. The error estimate for the alternating series test tells you that S is between 1/2 and 1 (in fact, it equals tex2html_wrap_inline413 , but we don't need that here), so that it's not zero. I'm going to add S to tex2html_wrap_inline417 as follows:


Summing, we get


where we have the same numbers as in the original series, except that they're rearranged so that the pattern of the signs is tex2html_wrap_inline419 instead of the original pattern tex2html_wrap_inline421 . But rearranging them changed the sum! (Since tex2html_wrap_inline423 , S is different from tex2html_wrap_inline427 .) Therefore rearranging the numbers in a conditionally convergent series can change its sum.

Tom Vogel
Mon May 5 12:53:33 CDT 1997