The limit of this sequence, if it exists, is called the value or limit of the infinite continued fraction with convergents , and is denoted by
In the same way, a finite sequence
defines a finite sequence
We then speak of a finite continued fraction with value .
It is not hard to prove that any irrational number is the value of a unique infinite simple continued fraction. Moreover, if denotes its th convergent, then is an alternating sequence and is decreasing (as well as convergent to zero). Also, the value of an infinite simple continued fraction is perforce irrational.
Any rational number is the value of two and only two finite continued fractions; in one of them, the last denominator is 1. E.g.
These two conditions on a real number are equivalent:
2. is irrational and its simple continued fraction is “eventually periodic”; i.e.
and, for some integer and some integer , we have for all .
and so on. If , we therefore expect
which indeed can be proved. As an exercise, you might like to look for a continued fraction expansion of the other solution of .
Although is transcendental, there is a surprising pattern in its simple continued fraction expansion.
No pattern is apparent in the expansions some other well-known transcendental constants, such as and Apéry’s constant .
where is a nonsquare integer . It turns out that if is any solution of the Pell equation other than , then is a convergent to .
and are well-known rational approximations to , and indeed both are convergents to :
For one more example, the distribution of leap years in the 4800-month cycle of the Gregorian calendar can be interpreted (loosely speaking) in terms of the continued fraction expansion of the number of days in a solar year.
|Date of creation||2013-03-22 12:47:12|
|Last modified on||2013-03-22 12:47:12|
|Last modified by||PrimeFan (13766)|
|Defines||simple continued fraction|