1 − 2 + 4 − 8 + ⋯

In mathematics, 1 − 2 + 4 − 8 + ⋯ is the infinite series whose terms are the successive powers of two with alternating signs.

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 quantityLeibniz did not quite assert that the series had a sum, but he did infer an association with ⁠1/3⁠ following Mercator's method.

[1][2] 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.

Briefly, if one expresses a partial sum of this series as a function of the penultimate term, one obtains either ⁠4m + 1/3⁠ or ⁠−4n + 1/3⁠.

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.

[6] His ideas on infinite series do not quite follow the modern approach; today one says that 1 − 2 + 4 − 8 + ... is Euler-summable and that its Euler sum is ⁠1/3⁠.