What is $1 - 1 + 1 - 1 + \cdots$? Grouped one way, $(1 - 1) + (1 - 1) + \cdots = 0$. Grouped another, $1 - (1 - 1) - (1 - 1) - \cdots = 1$. Euler and Leibniz thought the “right” answer was $\frac{1}{2}$, the average. In the eighteenth century such sums were handled freely, and the habit sometimes led to absurd results. Partial sums In chapter 6 of the Cours d’analyse, Cauchy cut the problem off at the root. A series $u_0 + u_1 + u_2 + \cdots$ has partial sums