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I would like to understand convergence with this example.

First let's describe two functions:

  • f(n): gives the nth element of an absolutely convergent infinite series.
  • g(n): gives the nth element of a conditional convergent infinite series.

let's redefine f(n) like this:

$$ f(n) = f(n) + g(n) - g(n) $$ $$ \sum_{i=0}^{i=\infty}{f(i)} = \sum_{i=0}^{i=\infty}{(f(i)+g(i)-g(i))} = \sum_{i=0}^{i=\infty}{f(i)} + \sum_{i=0}^{i=\infty}{g(i)} -\sum_{i=0}^{i=\infty}{g(i)} $$

Using the Riemann series theorem, this redefinition of f(n) let us reorder both "g(i) sigma terms" and the absolutely convergent series would converge to any value.

What's wrong with this redefinition of f(n)? Thanks a lot for your comments.

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    The point is that reordering the $g(i)$ terms changes the value of the series. So, the equality you have written is correct, but it ceases to be correct the moment that you reorder either $g(i)$ sum. – 2017-01-31
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    My interpretation of this theorem is that the result of conditional convergent infinite series is dependent of the process (algorithm) used to compute that mathematical object. I understand your point of view since you infer from the "sigma term" the "common order" of computing the series (f(1) + f(2) + f(3)...) but I think, nothing in principle (sum is commutative and associative) stops me to compute the sigma with another algorithm reordering the "normal order". – 2017-01-31

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