The triumphs and tribulations of summing a series

For a convergent series, the notion of its sum has never been in dispute. As the previous article recalled, starting with a sequence of real numbers (*u_n)_n *_{≥ 0} and denoting by (S*_n) _n *_{≥ 0} the associated sequence of partial sums, where S*_n = u*₀ + u₁ + u₂ +… + *u_n, we say that the series Σ u_n converges if the sequence (S_n)_n *_{≥ 0} converges to a limit S, called the sum of the series.

From the earliest investigations of series, however, some scholars sought to assign sums to non-convergent series by introducing definitions that generalized the properties of convergent series. One subject of debate, for example, was how to find a sum for the obviously divergent Grandi series, Σ (‒1)*ⁿ*. Grandi himself, followed by Leibniz and Euler, assigned it the sum 1/2 using various methods (see the In Brief article "A short history of Grandi's series").