The word "convergent" has a number of different meanings in mathematics.
Most commonly, it is an adjective used to describe a convergent sequence or convergent series, where it essentially means that the respective series or sequence approaches some limit (D'Angelo and West 2000, p. 259).
The rational number obtained by keeping only a limited number of terms in a continued fraction is also called a convergent. For example, in the simple continued fraction for the golden ratio,
 |
(1)
|
the convergents are
 |
(2)
|
Convergents are commonly denoted 
, 
, 
(ratios of integers), or 
(a rational number).
Given a simple continued fraction ![[b_0;b_1,b_2,...]](/images/equations/Convergent/Inline5.svg)
, the 
th convergent is given by the following ratio of tridiagonal
matrix determinants:
 |
(3)
|
For example, the third convergent of ![pi=[3;7,15]](/images/equations/Convergent/Inline7.svg)
is
 |
(4)
|
In the Wolfram Language, Convergents[terms] gives a list of the convergents corresponding to the specified list of continued
fraction terms, while Convergents[x,
n] gives the first 
convergents for a number 
.
Consider the convergents 
of a simple continued fraction ![[b_0;b_1,b_2,...]](/images/equations/Convergent/Inline11.svg)
, and define
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(5)
|
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(6)
|
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(7)
|
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(8)
|
Then subsequent terms can be calculated from the recurrence relations
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(9)
|
 |  |  |
(10)
|
, 2, ..., 
.
For a generalized continued fraction 
, the recurrence
generalizes to
 |  |  |
(11)
|
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(12)
|
The continued fraction fundamental recurrence relation for a simple continued fraction is
 |
(13)
|
It is also true that if 
,
 |  | ![[b_n;b_(n-1),...,b_0]](/images/equations/Convergent/Inline42.svg) |
(14)
|
 |  | ![[b_n;b_(n-1),...,b_1].](/images/equations/Convergent/Inline45.svg) |
(15)
|
Furthermore,
 |
(16)
|
Also, if a convergent 
,
then
![(B_n)/(A_n)=[0;b_0,b_1,...,b_n].](/images/equations/Convergent/NumberedEquation7.svg) |
(17)
|
Similarly, if 
,
then 
and
![(B_n)/(A_n)=[0;b_1,...,b_n].](/images/equations/Convergent/NumberedEquation8.svg) |
(18)
|
The convergents 
also satisfy
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(19)
|
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(20)
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Plotted above on semilog scales are 
(
even; left figure) and 
(
odd; right figure) as a function of 
for the convergents of 
. In general, the even convergents
of an infinite simple continued
fraction for a number 
form an increasing sequence, and the odd
convergents 
form a decreasing sequence (so any even
convergent is less than any odd convergent). Summarizing,
 |
(21)
|
 |
(22)
|
Furthermore, each convergent for 
lies between the two preceding ones. Each convergent
is nearer to the value of the infinite continued fraction than the previous one.
In addition, for a number ![x=[b_0;b_1,b_2,...]](/images/equations/Convergent/Inline66.svg)
,
 |
(23)
|
In the course of searching for continued fraction identities, Raayoni (2021) and Elimelech et al. (2023) noticed that while
the numerator and denominator of convergents 
in general grow factorially, the reduced numerator and
denominator 
and 
for 
grow at most exponentially, i.e., as 
. They termed this phenomenon factorial
reduction and noted that while it is extremely rare in general, it holds for
all identities originally found by the Ramanujan Machine (Raayoni et al.
2021).
