rearrangement theorem

For ordinary finite addition, the order of the summands never matters — a + b + c equals c + a + b. The rearrangement theorem reveals that this comforting law breaks down for infinite series unless convergence is absolute. Reorder a conditionally convergent series and you can steer its sum to wherever you like.

Riemann's theorem states it sharply: if sum a_n is a conditionally convergent series of real numbers, then for any target value S (including +infinity or -infinity) there is a rearrangement — a bijection of the index set with itself — for which the reordered series converges to S, or one that makes it diverge or oscillate. The mechanism is that the positive part and the negative part each sum to infinity, so one can greedily pile up positives until overshooting the target, then negatives until undershooting, repeating forever.

The companion positive result is just as important: if sum a_n converges absolutely, then every rearrangement converges to the same sum. So absolute convergence is precisely the condition under which the order of summation is irrelevant. The theorem is a vivid warning that the algebra of finite sums must not be applied blindly to infinite ones.