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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 A_n/B_n , p_n/q_n , P_n/Q_n (ratios of integers), or c_n (a rational number).

Given a simple continued fraction [b_0;b_1,b_2,...] , the th convergent is given by the following ratio of tridiagonal matrix determinants:

| $(A_n)/(B_n)=(\|b_0 -1 0 ... 0; 1 b_1 -1 ... 0; 0 1 b_2 ... 0; 0 0 1 ... -1; 0 0 0 ... b_n|)/(|b_1 -1 ... 0; 1 b_2 ... 0; 0 1 ... -1; 0 0 ... b_n|).$ | (3) | | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --- |

For example, the third convergent of pi=[3;7,15] is

| $(A_3)/(B_3)=(\|3 -1 0; 1 7 -1; 0 1 15|)/(|7 -1; 1 15|)=(333)/(106).$ | (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 c_n=A_n/B_n of a simple continued fraction [b_0;b_1,b_2,...] , and define

Then subsequent terms can be calculated from the recurrence relations

k=1 , 2, ..., .

For a generalized continued fraction K_(k=1)^(infty)a_k/b_k , the recurrence generalizes to

The continued fraction fundamental recurrence relation for a simple continued fraction is

	(13)

It is also true that if b_0!=0 ,

Furthermore,

	(16)

Also, if a convergent c_n=A_n/B_n>1 , then

	(17)

Similarly, if c_n=A_n/B_n<1 , then b_0=0 and

	(18)

The convergents A_n/B_n also satisfy

CFConvergents

Plotted above on semilog scales are c_n-pi ( even; left figure) and pi-c_n ( odd; right figure) as a function of for the convergents of . In general, the even convergents c_(2n) of an infinite simple continued fraction for a number form an increasing sequence, and the odd convergents c_(2n+1) form a decreasing sequence (so any even convergent is less than any odd convergent). Summarizing,

	(21)

	(22)

Furthermore, each convergent for n>=3 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,...] ,

| $1/((b_(n+1)+2)B_n^2)<\|x-(A_n)/(B_n)|<1/(b_(n+1)B_n^2).$ | (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 p_n/q_n in general grow factorially, the reduced numerator and denominator p_n/g_n and q_n/g_n for g_n=GCD(p_n,q_n) grow at most exponentially, i.e., as s^n . 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).

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References

D'Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.Elimelech, R.; David, O.; De la Cruz Mengual, C.; Kalisch, R.; Berndt, W.; Shalyt, M.; Silberstein, M.; Hadad, Y.; and Kaminer, I. "Algorithm-Assisted Discovery of an Intrinsic Order Among Mathematical Constants." 22 Aug 2023.https://arxiv.org/abs/2308.11829.Liberman, H. Simple Continued Fractions: An Elementary to Research Level Approach. SMD Stock Analysts, pp. II-9-II-10, 2003.Raayoni, G; Gottlieb, S.; Manor, Y.; Pisha, G.; Harris, Y.; Mendlovic, U.; Haviv, D.; Hadad, Y.; and Kaminer, I. "Generating Conjectures on Fundamental Constants With the Ramanujan Machine."Nature 590, 67-73, 2021.

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Convergent

Cite this as:

Weisstein, Eric W. "Convergent." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Convergent.html

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