The Official SAT Question of the Day. Nerd Paradise. 1 − 2 + 3 − 4 + · · · The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ...
The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation: A rigorous explanation of this equation would not arrive until much later. 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 easily assign to 1 − 2 + 3 − 4 + ... a "sum" of 1⁄4 after all.