1 − 2 + 4 − 8 + ⋯ a convergent geometric series whose sum is by the usual formula. The Borel sum of is also ; when Émile Borel introduced the limit formulation of Borel summation in 1896, this was one of his first examples after 1 − 1 + 1 − 1 + ⋯ 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
1/2 + 1/4 + 1/8 + 1/16 + ⋯ 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 infinite series, the infinite sum is defined to mean the limit of the sum of the first terms as approaches infinity. Multiplying by 2 reveals a useful relationship: Subtracting from both sides, As approaches