1 − 2 + 4 − 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ In mathematics, 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 . 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
1/2 + 1/4 + 1/8 + 1/16 + ⋯ meters, and so on Zeno argued that the tortoise would always remain ahead of Achilles. The parts of the Eye of Horus were once thought to represent the first six summands of the series. 1/2 + 1/4 + 1/8 + 1/16 + ⋯ In mathematics, the infinite series is an elementary example of a geometric series that converges absolutely. There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2 + 2 + 2 + ... The sum of this series can be denoted in summation notation as: As with any