# 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.