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The Reagan administration made Letter-size paper the norm for US federal forms in the early 1980s; previously, the smaller "official" Government Letter size, 8 by 10.5 inches (203.2 by 266.7 mm) (aspect ratio: 1.3125), was used in government, while 8.5-by-11-inch (215.9 by 279.4 mm) paper was standard in most other offices. [2]
It has the height of Canadian P4 paper (215 mm × 280 mm, about 8 + 1 ⁄ 2 in × 11 in) and the width of international A4 paper (210 mm × 297 mm or 8.27 in × 11.69 in), i.e. it uses the smaller value among the two for each side. The table shows how this format can be generalized into an entire format series.
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
For example, a quarto (from Latin quartō, ablative form of quartus, fourth [3]) historically was a book printed on sheets of paper folded in half twice, with the first fold at right angles to the second, to produce 4 leaves (or 8 pages), each leaf one fourth the size of the original sheet printed – note that a leaf refers to the single piece ...
6 1 2 1 1 −1 4 5 9. and would be written in modern notation as 6 1 / 4 , 1 1 / 5 , and 2 − 1 / 9 (i.e., 1 8 / 9 ). The horizontal fraction bar is first attested in the work of Al-Hassār (fl. 1200), [35] a Muslim mathematician from Fez, Morocco, who specialized in Islamic inheritance jurisprudence.
ª Approximate measure in current use in Latin America: 216 x 341 mm. Ring binders or lever arch files designed to hold foolscap folios are often used to hold A4 paper (210 × 297 mm, 8 + 1 ⁄ 4 × 11 + 3 ⁄ 4 in). The slightly larger size of such a binder offers greater protection to the edges of the pages it contains.
The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯.The parabola is their smoothed asymptote; its y-intercept is −1/12. [1]The infinite series whose terms ...
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1 / 4 + 1 / 16 + ⋯ are: + + + + = +. This form can be proved by multiplying both sides by 1 − 1 / 4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs.