This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Periodic continued fraction" – news · newspapers · books · scholar · JSTOR(January 2014) (Learn how and when to remove this message)
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
where the initial block [a0, a1,...ak] of k+1 partial denominators is followed by a block [ak+1, ak+2,...ak+m] of m partial denominators that repeats ad infinitum. For example, can be expanded to the periodic continued fraction [1,2,2,2,...].
This article considers only the case of periodic regular continued fractions. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers. The general case, where the partial denominators ai are arbitrary real or complex numbers, is treated in the article convergence problem.
and 24 Related for: Periodic continued fraction information
descriptions as a fallback PeriodiccontinuedfractionContinued Logarithms Complete quotient Computing continuedfractions of square roots – Algorithms...
when the elements of the continuedfraction are complex numbers. An infinite periodiccontinuedfraction is a continuedfraction of the form x = a 1 b 1...
partial denominators ai are less than or equal to M. A regular periodiccontinuedfraction consists of a finite initial block of partial denominators followed...
integer as a continuedfraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continuedfraction. In a sense...
{\displaystyle S_{c}} in a continuedfraction corresponds to reducing the quadratic form. The eventually periodic nature of the continuedfraction is then reflected...
quadratic irrational. Therefore, π cannot have a periodiccontinuedfraction. Although the simple continuedfraction for π (shown above) also does not exhibit...
\mathbb {Z^{+}} } . Since this is a quadratic irrational, the continuedfraction must be periodic (unless n is square, in which case the factorization is obvious)...
have seven neutrons, and a very small fraction have eight neutrons. Isotopes are never separated in the periodic table; they are always grouped together...
quadratic irrational x , {\displaystyle x,} the sequence becomes a periodiccontinuedfraction. The ordered set ( R > 0 , > ) {\displaystyle \left(\mathbb {R}...
is necessary, and not just convenient. Consider the infinite periodiccontinuedfraction f ( z ) = 1 + z 1 + z 1 + z 1 + z ⋱ . {\displaystyle f(z)=1+{\cfrac...
do not have apparent pattern in their continuedfraction expansions, since eventually periodiccontinuedfractions correspond to quadratic irrationals,...
decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever);...
Extended periodic table Element 119 (Uue, marked here) in period 8 (row 8) marks the start of theorisations. An extended periodic table theorizes about...
mass fraction (in commercial contexts often called weight fraction), by mole fraction (fraction of atoms by numerical count, or sometimes fraction of molecules...
many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The...
4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.} The continuedfraction representations can be derived from Euler's continuedfraction formula and express the real number...
which the columns ("groups") share recurring ("periodic") physical and chemical properties. The periodic table summarizes various properties of the elements...
( x ) . {\displaystyle {\frac {d}{dx}}e^{f(x)}=f'(x)e^{f(x)}.} A continuedfraction for ex can be obtained via an identity of Euler: e x = 1 + x 1 − x...
continue to work towards a university position. In 1825, he obtained the degree of Doctor of Philosophy with a dissertation on the partial fraction decomposition...
{\displaystyle \textstyle \alpha } , the continued-fraction expansion of α {\displaystyle \textstyle \alpha } is periodic if and only if α {\displaystyle \textstyle...
with hydrogen they constitute group 1, which lies in the s-block of the periodic table. All alkali metals have their outermost electron in an s-orbital:...
Global Information
