Riemann series theorem

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In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent. As an example, the series 1 − 1 + 1 2 − 1 2 + 1 3 − 1 3 + 1 4 −...

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