A conditionally convergent real series can be rearranged to converge to any prescribed real value or to diverge. Its positive and negative subseries have infinite total magnitudes: successively add positive terms to cross a desired value and negative terms to cross back. Since the terms tend to zero, the overshoots shrink to zero.
A rearrangement changes the order of a series without changing its multiset of terms. A conditionally convergent real series can be rearranged to approach any prescribed real limit or to diverge.
The Riemann series theorem specifies the possible sums of rearrangements of a conditionally convergent real series.
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The Riemann Series Theorem, also known as the Riemann rearrangement theorem, is a result in the field of analysis concerning the summation of series of real or complex numbers. The theorem states that if a series is conditionally convergent, the terms of that series can be rearranged in such a way that the rearranged series converges to any real number, or even diverges.