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An example of molloby's idea appears on page 77 of Goldberg's Methods of Real Analysis (second edition):
We have that
(1) \sum_{n=1}^\infty (-1)^{n+1}/n = L = \log 2. (where L is the sum of the series).
Now, divide L by 2, giving
L/2 = 1/2 - 1/4 + 1/6 - 1/8 + ..., and thus
(2) L/2 = 0 + 1/2 - 0 - 1/4 + 0 + 1/6 - 0 - 1/8 + ... .
Adding (2) and (1) gives
(3) 3/2 L = 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + ... .
You will notice that (3) is just a rearrangement of (1), yet it sums to 3/2 L = 3/2 \log 2.
The book goes on to give the following theorem:
Let L be a conditionally convergent series of real numbers. Then for any real number x, there is a rearrangement of L which converges to x. However, if the series is absolutely convergent, then all rearrangements sum to the same value.
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