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In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
Slices of approximately 1/8 of a pizza. A unit fraction is a positive fraction with one as its numerator, 1/ n.It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number.
The statement that 1 / 2 − 1 / 4 + 1 / 8 − 1 / 16 + ⋯ is absolutely convergent means that the series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ⋯ is convergent. In fact, the latter series converges to 1, and it proves that one of the binary expansions of 1 is 0.111....
Archimedes' figure with a = 3 / 4 In mathematics, the infinite series 1 / 4 + 1 / 16 + 1 / 64 + 1 / 256 + ⋯ is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC. [1]
In it, uniform blocks are stacked on top of each other to achieve the maximum sideways or lateral distance covered. The blocks are stacked 1/2, 1/4, 1/6, 1/8, 1/10, … distance sideways below the original block. This ensures that the center of gravity is just at the center of the structure so that it does not collapse.
The first four partial sums of 1 + 2 + 4 + 8 + ⋯. In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. 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 to infinity, so the sum of this series is infinity.
Until then, workers such as millwrights, boilermakers, and machinists in the Anglosphere measured only in traditional fractions of an inch, divided via successive halving, usually only as far as 64ths (1, 1 ⁄ 2, 1 ⁄ 4, 1 ⁄ 8, 1 ⁄ 16, 1 ⁄ 32, 1 ⁄ 64). Each 64th is about 16 thou.
where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]