Wikipedia:Today's featured article/April 15, 2007

The first few thousand terms and partial sums of the series

In mathematics, 1 − 2 + 3 − 4 + · · · is the infinite series whose terms are the successive positive integers, given alternating signs. The series diverges, meaning that its sequence of partial sums (1, −1, 2, −2, …) does not tend towards any finite limit. Nonetheless, Leonhard Euler claimed that 1 − 2 + 3 − 4 + · · · = ¹⁄₄. Starting in 1890, Ernesto Cesàro, Émile Borel, and others investigated well-defined methods to assign generalized sums to divergent series – including new interpretations of Euler's attempts. Many of these summability methods assign to 1 − 2 + 3 − 4 + · · · a "sum" of ¹⁄₄ after all. Cesàro summation is one of the few methods that does not sum 1 − 2 + 3 − 4 + · · ·, so the series is an example where a slightly stronger method, such as Abel summation, is required. The series 1 − 2 + 3 − 4 + · · · is closely related to Grandi's series 1 − 1 + 1 − 1 + · · ·. Euler treated these two as special cases of 1 − 2ⁿ + 3ⁿ − 4ⁿ + · · · for arbitrary n, a line of research extending his work on the Basel problem and leading towards the functional equations of what we now know as the Dirichlet eta function and the Riemann zeta function. (more...)

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