# Footnote 3

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The series giving the gain, ie A2, may turn out to be a conditionally convergent series, similar to the alternating harmonic series:

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 ...

This series sums to ln2, but only if we sum it in the order given above. This and other conditionally convergent series can be re-arranged to sum to any value we care to name. We can split the terms to get

ln2 = (2 - 1) - 1/2 + (2/3 - 1/3) - 1/4 + (2/5 - 1/5) - 1/6 + (2/7 - 1/7) ...

and divide both sides by 2, giving

ln2/2 = 1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 + 1/7 - 1/14 ...

The right hand side of the above equation contains exactly the same terms as the original, yet the sum is half the value due to re-arrangement. Note that we have not changed the order of summation in the above steps so that the logic is valid.

Clark and Shackel show an instance of the paradox where the gain is a conditionally convergent series similar to the above. In other words they give an instance of A2 that converges to a non-zero value. How do we arrive at A2? It is by summing the gain in the most obvious way, ie by adding the positive term in the first pair of envelopes, subtracting the first negative, then adding the positive term in the second pair and subtracting the negative term in the second pair, and so on.

Alternatively, we can sum the gain by adding the first positive amount, then subtract the negative term from the same pair, then subtract the negative term from the second pair. Then we add the positive term from the second pair, subtract the negative terms from the third and fourth pairs. And so on, mirroring the procedure that sums to ln2/2 in the above example. Obviously, by summing in this way we will reach a different value for the gain if A2 is conditionally convergent. Since the gain calculation must give us a single value it follows that we cannot sum A2 to calculate the gain.

It may be objected that we are summing the gain in an "unnatural way". While it is true that the alternative summation given above is not the obvious way to sum the gain, there is nothing to say that it is "wrong". It is just as valid to sum it in this order as the "natural" or canonical order that gives A2.