is zero. Leibniz thought that the argument from was valid; he took it as an example of his law of continuity. Since the relation holds for all x less than 1, it should hold for x equal to 1 as well. Still, Leibniz thought that one should be able to find the sum of the series directly, without needing to refer back to the expression from which it came. This approach may seem obvious by modern standards, but it is a significant step from the point of view of the history of summing divergent series. In the 18th century, the study of series was dominated by power series, and summing a numerical series by expressing it as f(1) of some function's power series was thought to be the most natural strategy. Leibniz begins by observing that taking an even number of terms from the series, the last term is −1 and the sum is 0: 1 − 1 = 1 − 1 + 1 − 1 = 1 − 1 + 1 − 1 + 1 − 1 = 0. Taking an odd number of terms, the last term is +1 and the sum is 1: 1 = 1 − 1 + 1 = 1 − 1 + 1 − 1 + 1 = 1. Now, the infinite series 1 − 1 + 1 − 1 + · · · has neither an even nor an odd number of terms, so it produces neither 0 nor 1; by taking the series out to infinity, it becomes something between those two options. There is no more reason why the series should take one value than the other, so the theory of "probability" and the "law of justice" dictate that one should take the arithmetic mean of 0 and 1, which is Eli Maor says of this solution, "Such a brazen, careless reasoning indeed seems incredible to us today…" Kline portrays Leibniz as more self-conscious: "Leibniz conceded that his argument was more metaphysical than mathematical, but said that there is more metaphysical truth in mathematics than is generally recognized." Charles Moore muses that Leibniz would hardly have had such confidence in his metaphysical strategy if it did not give the same result (namely 1⁄2) as other approaches. Mathematically, this was no accident: Leibniz's treatment would be partially justified when the compatibility of averaging techniques and power series was finally proven in 1880.