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Footnote 1

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What does that mean? The sum is not meaningful because it oscillates. Since there is no last term in an infinite series we cannot say whether it is odd or even, ie whether it is positive or negative, and hence whether the sum is positive or zero. Increasing oscillating series like this have a further interesting property, namely that by re-arranging their terms we can make them appear to sum to widely different totals. Here is an illustration:

(a) 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 + ... = ( 1 - 1 ) + ( 2 - 2 ) + ( 3 - 3 ) + ( 4 - 4 ) + ...

                                                   = 0 + 0 + 0 + 0 + ...                            = 0

(b) 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 + ... = 1 + ( - 1 + 2 ) + ( - 2 + 3 ) + ( - 3 + 4 ) + ( - 4 + 5 ) + ...

                                                   = 1 + 1 + 1 + 1 + 1 + ...                      = ∞

(c) 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 + ... = ( 1 - 1 ) + ( 2 - 2 ) + ( 3 - 3 ) + ( 4 - 4 ) + ...

                                                   = - 1 + 1 - 2 + 2 - 3 + 3 - 4 + 4 + ... (reversing the order in each pair)

                                                   = - 1 + ( 1 - 2 ) + ( 2 - 3 ) + ( 3 - 4 ) + ( 4 - 5 ) + ...

                                                   = - 1 - 1 - 1 - 1 - 1 + ...                        = -∞

The point is that a series like the one above simply cannot be summed, and that we can manipulate it to appear to sum to just about any value. The same is true of the series S, that sums the gain from our distribution.

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