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In mathematics, 1 − 2 + 4 − 8 + ... is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. : As a series of real numbers it diverges, so in the usual sense it has no sum. In a much broader sense, the series has a generalized sum of ⅓. ==Historical arguments== Gottfried Leibniz considered the divergent alternating series as early as 1673. He argued that by subtracting either on the left or on the right, one could produce either positive or negative infinity, and therefore both answers are wrong and the whole should be finite: :"Now normally nature chooses the middle if neither of the two is permitted, or rather if it cannot be determined which of the two is permitted, and the whole is equal to a finite quantity." Leibniz did not quite assert that the series had a ''sum'', but he did infer an association with ⅓ following Mercator's method.〔Leibniz pp.205-207; Knobloch pp. 124–125. The quotation is from ''De progressionibus intervallorum tangentium a vertice'', in the original Latin: "Nunc fere cum neutrum liceat, aut potius cum non possit determinari utrum liceat, natura medium eligit, et totum aequatur finito."〕 The attitude that a series could equal some finite quantity without actually adding up to it as a sum would be commonplace in the 18th century, although no distinction is made in modern mathematics.〔Ferraro and Panza p.21〕 After Christian Wolff read Leibniz's treatment of Grandi's series in mid-1712,〔Wolff's first reference to the letter published in the ''Acta Eruditorum'' appears in a letter written from Halle, Saxony-Anhalt dated 12 June 1712; Gerhardt pp. 143–146.〕 Wolff was so pleased with the solution that he sought to extend the arithmetic mean method to more divergent series such as . Briefly, if one expresses a partial sum of this series as a function of the penultimate term, one obtains either or . The mean of these values is , and assuming that at infinity yields ⅓ as the value of the series. Leibniz's intuition prevented him from straining his solution this far, and he wrote back that Wolff's idea was interesting but invalid for several reasons. The arithmetic means of neighboring partial sums do not converge to any particular value, and for all finite cases one has , not . Generally, the terms of a summable series should decrease to zero; even could be expressed as a limit of such series. Leibniz counsels Wolff to reconsider so that he "might produce something worthy of science and himself."〔The quotation is Moore's (pp. 2–3) interpretation; Leibniz's letter is in Gerhardt pp.147-148, dated 13 July 1712 from Hanover.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「1 − 2 + 4 − 8 + ⋯」の詳細全文を読む スポンサード リンク
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