Riemann series theorem [SEP] named after
Riemann series theorem
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, or diverges.
As an example, the series 1 – 1 + 1/2 – 1/2 + 1/3 – 1/3 + ... converges to 0 (for a sufficiently large number of terms, the
considerable amount of work on "Summability" and convergence properties of infinite series. In fact, a theorem named after Rau figures in Hardy's book. Rau published several papers, primarily in three fields: summability of series, theory of functions of a complex variable and sums of an even number of squares. Among the first papers he wrote after his return was on the subject of the Riemann zeta function, and the technique devised by Rau was found to have wide applicability in other problems of number theory.
Teaching influence