The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that
For instance, Euler's number e has the Engel expansion[1]
1, 1, 2, 3, 4, 5, 6, 7, 8, ...
corresponding to the infinite series
Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. If x is rational, its Engel expansion provides a representation of x as an Egyptian fraction. Engel expansions are named after Friedrich Engel, who studied them in 1913.
An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion.
^Sloane, N. J. A. (ed.). "Sequence A028310". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
The Engelexpansion of a positive real number x is the unique non-decreasing sequence of positive integers ( a 1 , a 2 , a 3 , … ) {\displaystyle (a_{1}...
nondecreasing. Every rational number has a finite Engelexpansion, while irrational numbers have an infinite Engelexpansion. Anshel & Goldfeld (1991) study numbers...
2}+{1 \over 6}+{1 \over 30}+\ldots =0{.}7052301717918\ldots } The Engelexpansion of this number results in the sequence of the prime numbers (See (sequence...
Melancholie der Engel (English: The Angels' Melancholia) is a 2009 German independent arthouse horror film directed, shot and edited by Marian Dora and...
square roots Egyptian fraction – Finite sum of distinct unit fractions Engelexpansion – decomposition of a positive real number into a series of unit fractions...
this way include Engelexpansion, in which each successive denominator must be a multiple of the previous one, and odd greedy expansion, in which all denominators...
original on 16 May 2008. Retrieved 11 May 2022. Magone 2009, pp. 385–386. Engel, Jeffrey A. (2009). The Fall of the Berlin Wall: The Revolutionary Legacy...
Kommunistischen Partei), is a political pamphlet written by Karl Marx and Friedrich Engels, commissioned by the Communist League and originally published in London...
x^{-1}\right\rfloor \right)-1} is iterated to find the coefficients appearing in Engelexpansions. The Mellin transform of the map g ( x ) {\displaystyle g(x)} is related...
for the 84th Congress and the 85th Congress. He sponsored several major expansions of the California Central Valley Project as well as the Saline Water Conversion...
Chemistry, 8th ed. Oxford University Press. p. 79. ISBN 978-0-19-870072-2. Engel, Thomas; Philip Reid (2006). Physical Chemistry. Pearson Benjamin Cummings...
meaning in the early 20th century to primarily refer to European imperial expansion and the imperial subjection of Asian and African peoples. Collins English...
Journal of Monetary Economics, 52(2), pp. 281–305. Devereux, Michael B.; Engel, Charles (2003). "Monetary Policy in the Open Economy Revisited: Price Setting...
reasons, reasons we fail to appreciate if we collapse them all together."; Engel 2021, pp. 3 ("This book is about an encounter between two sets of human...
Mark Cendrowski Story by : Bill Prady & Chuck Lorre Teleplay by : Stephen Engel & Tim Doyle October 13, 2008 (2008-10-13) 3T7354 9.34 Raj is included in...
Housing Question" is an essay by Friedrich Engels that was published in 1872. Engels wrote: The expansion of the big modern cities gives the land in certain...
2007 as a project of the Center for the Expansion of Fundamental Rights. In 2012, the Center for the Expansion of Fundamental Rights officially changed...
make their livelihoods. For Marx and his lifetime collaborator, Friedrich Engels, historical materialism is the "view of the course of history which seeks...
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