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1/2 − 1/4 + 1/8 − 1/16 + ⋯. Demonstration that 1 2 − 1 4 + 1 8 − 1 16 + ⋯ = 1 3. In mathematics, the infinite series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely . It is a geometric series whose first term is 1 2 and whose common ratio is − 1 2, so its sum is.
The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.
1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. 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 ...
1,3-Propane sultone is the organosulfur compound with the formula (CH 2) 3 SO 3. It is a cyclic sulfonate ester, a class of compounds called sultones . [ 2 ] [ 3 ] It is a readily melting colorless solid.
4 3 2 1 is a novel by Paul Auster published in January 2017. At the time of its publication, it was the first new Auster novel to have appeared in seven years. [ 1][ 2] Auster worked on the book seven days a week for three-and-a-half-years and wrote it in long hand. At 866 pages, the novel is much longer than any of his previous works. [ 3]
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. However, it can be manipulated to yield a number of ...
1,3-Dipolar cycloaddition. The 1,3-dipolar cycloaddition is a chemical reaction between a 1,3-dipole and a dipolarophile to form a five-membered ring. The earliest 1,3-dipolar cycloadditions were described in the late 19th century to the early 20th century, following the discovery of 1,3-dipoles.
For example, raltegravir is an HIV drug which contains an 1,3,4-oxadiazole ring. Other pharmaceutical drugs containing the 1,3,4-oxadiazole ring include fenadiazole, zibotentan, and tiodazosin. 1,3,4-Oxadiazole derivatives can be synthesized in a variety of ways. [3] One pathway is from oxidation of tetrazoles in the presence of aldehydes. [4]