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Pairing up the terms of the series 1 / 2 − 1 / 4 + 1 / 8 − 1 / 16 + ⋯ results in another geometric series with the same sum, 1 / 4 + 1 / 16 + 1 / 64 + 1 / 256 + ⋯. This series is one of the first to be summed in the history of mathematics ; it was used by Archimedes circa 250–200 BC.
For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term −1/12, and the negative ...
Paul Mescal revealed to Vanity Fair as part of the first look at “Gladiator II” that he was offered the sequel’s leading role after meeting director Ridley Scott for only a half-hour. The ...
The street-style sighting, which occurred Thursday in New York City, involved a blue T-shirt that read in large white letters: “I'm Retired.”. Okay, Rihanna, we can take a hint! Below that ...
Roku’s select Series 4K TVs are built to bring a high-quality streaming experience to viewers. With built-in sound features like auto speech clarity and night mode, this Roku TV will deliver ...
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The geometric series on the real line. 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. The series is related to philosophical questions considered in antiquity, particularly ...