Download
1 / 30
E N D
1. Session4.Infinite Sequences Approximation of Real Numbers,
Continued Fractions,
and Limit of Infinite Sequences.
2. Limit of a sequence n nsin(1/n)
1 0.841471
2 0.958851
...
10 0.998334
...
100 0.999983
As the positive integer n becomes larger and larger, the value nsin(1/n) becomes arbitrarily close to 1. We say that "the limit of the sequence nsin(1/n) equals1.
The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit.
Intuitively, suppose we have a sequence of points (i.e. an infinite set of points labelled using the natural numbers) in some sort of mathematical object (for example the real numbers or a vector space) which has a concept of nearness (such as "all points within a given distance of a fixed point").
A point L is the limit of the sequence if for any prescribed nearness, all but a finite number of points in the sequence are that near to L.
This may be visualised as a set of spheres of size decreasing to zero, all with the same centre L, and for any one of these spheres, only a finite number of points in the sequence being outside the sphere.
3. Continued fraction In mathematics, a continued fraction is an expression such as
where a0 is an integer and all the other numbers ai (i ? 0) are positive integers. Longer expressions are defined analogously. If the partial numerators and partial denominators are allowed to assume arbitrary values, which may in some contexts include functions, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the standard form above from generalized continued fractions, it may be called a simple or regular continued fraction, or is said to be in canonical form.
4. Motivation for Continued Fractions The study of continued fractions is motivated by a desire to have a "mathematically pure" representation for the real numbers.
Most people are familiar with the decimal representation of real numbers, which may be defined by
where a0 may be any integer, and every other ai is an element of {0, 1, 2, ..., 9}. In this representation, the number p, for example, is represented by the sequence of integers (ai) = (3, 1, 4, 1, 5, 9, 2, ...).
This decimal representation has some problems. One problem is that many rational numbers lack finite representations in this system. For example, the number 1/3 is represented by the infinite sequence (0, 3, 3, 3, 3, ....). Another problem is that the constant 10 is an essentially arbitrary choice, and one which biases the resulting representation toward numbers that have some relation to the integer 10. For example, 137/1600 has a finite decimal representation, while 1/3 does not, not because 137/1600 is simpler than 1/3, but because 1600 happens to divide a power of 10 (106 = 1600 625). Continued fraction notation is a representation of the real numbers that avoids both these problems.
5. Continued Fractions by Approximation Let us consider how we might describe a number like 415/93, which is around 4.4624. This is approximately 4. Actually it is a little bit more than 4, about 4 + 1/2. But the 2 in the denominator is not correct; the correct denominator is a little bit more than 2, about 2 + 1/6, so 415/93 is approximately 4 + 1/(2 + 1/6). But the 6 in the denominator is not correct; the correct denominator is a little bit more than 6, actually 6+1/7. So 415/93 is actually 4+1/(2+1/(6+1/7)). This is exact.
Dropping the redundant parts of the expression 4+1/(2+1/(6+1/7)) gives the abbreviated notation [4; 2, 6, 7].
The continued fraction representation of real numbers can be defined in this way.
6. Continued fraction representation has several desirable properties:
The continued fraction representation for a number is finite if and only if the number is rational.
Continued fraction representations for "simple" rational numbers are usually short.
Every rational number has an essentially unique continued fraction representation. Each rational can be represented in exactly two ways, since [a0; a1, ... an, 1] = [a0; a1, ... an+1]. Mathematicians conventionally choose the shorter representation.
The continued fraction representation of an irrational number is unique.
The terms of a continued fraction will repeat if and only if it is the continued fraction representation of a quadratic irrational, that is, a real solution to a quadratic equation with integer coefficients. For example, the repeating continued fraction [1; 1, 1, 1, ...] is the golden ratio, and the repeating continued fraction [1; 2, 2, 2, ...] is the square root of 2.
Truncating the continued fraction representation of a number x early yields a rational approximation for x which is in a certain sense the "best possible" rational approximation (see theorem 5, corollary 1 below for a formal statement).
7. Truncation This last property is extremely important, and is not true of the conventional decimal representation. Truncating the decimal representation of a number yields a rational approximation of that number, but not usually a very good approximation. For example, truncating 1/7 = 0.142857... at various places yields approximations such as 142/1000, 14/100, and 1/10. But clearly the best rational approximation is "1/7" itself. Truncating the decimal representation of p yields approximations such as 31415/10000 and 314/100. The continued fraction representation of p begins [3; 7, 15, 1, 292, ...]. Truncating this representation yields the excellent rational approximations 3, 22/7, 333/106, 355/113, 103993/33102, ... The denominators of 314/100 and 333/106 are almost the same, but the error in the approximation 314/100 is nineteen times as large as the error in 333/106. As an approximation to p, [3; 7, 15, 1] is more than one hundred times more accurate than 3.1416.
8. The integers a0,a1, etc., are called the quotients of the continued fraction. One can abbreviate a continued fraction as
or, in the notation of Pringsheim, as
Here is another related notation:
Sometimes angle brackets are used, like this:
The semicolon in the square and angle bracket notations is sometimes replaced by a comma.
One may also define infinite simple continued fractions as limits:
This limit exists for any choice of positive integers a0,a1,....
9. Finite continued fractions Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore greater than 1, if the short representation has at least two terms. In symbols:
10. Infinite continued fractions Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction.
An infinite continued fraction representation for an irrational number is mainly useful because its initial segments provide excellent rational approximations to the number. These rational numbers are called the convergents of the continued fraction. Even-numbered convergents are smaller than the original number, while odd-numbered ones are bigger.
For a continued fraction [a0;a1,a2,...], the first four convergents (numbered 0 through 3) are
In words, the numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent. The denominators are formed similarly.
11. Periodic continued fractions The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients (rational solutions have finite continued fraction expansions as previously stated). The simplest examples are the golden ratio f = [1; 1, 1, 1, 1, 1, ...] and v 2 = [1; 2, 2, 2, 2, ...]; while v14 = [3;1,2,1,6,1,2,1,6...] and v42 = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for v 2) or 1,2,1 (for v14), followed by the double of the leading integer.
12. A property of the golden ratio f An interesting result, stemming from the fact that the continued fraction expansion for f doesn't use any integers greater than 1, is that f is one of the most "difficult" real numbers to approximate with rational numbers. One theorem states that any real number k can be approximated by rational m/n with
While virtually all real numbers k will eventually have infinitely many convergents m/n whose distance from k is significantly smaller than this limit, the convergents for f (i.e., the numbers 5/3, 8/5, 13/8, 21/13, etc.) consistently "toe the boundary", keeping a distance of almost exactly away from f, thus never producing an approximation nearly as impressive as, for example, 355/113 for p. It can also be shown that every real number of the form (a+bf)/(c+df) where a, b, c, and d are integers such that ad-bc=1 shares this property with the golden ratio f.
13. Regular patterns in continued fractions While one cannot discern any pattern in the simple continued fraction expansion of p, this is not true for e, the base of the natural logarithm:
We also have, when n is an integer greater than one,
Another, more complex pattern appears in this continued fraction expansion, where n is odd:
Other continued fractions of this sort are
and, for integral n>1,
14. Formal definition of limit of numbers For a sequence of real numbers
A real number L is said to be the limit of the sequence xn, written
if and only if for every real number e>0, there exists a natural number N such that for every n>N we have |xn-L|<e.
For a sequence of points in a metric space M with distance function d (such as a sequence of rational numbers, real numbers, complex numbers, points in a normed space, etc.):
An element is said to be the the limit of the sequence, written
if and only if for every real number e>0, there exists a natural number N such that for every n>N, we have d(xn,L)<e.
15. Examples The sequence 1, -1, 1, -1, 1, ... is divergent.
The sequence 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ... converges with limit 1. This is an example of an infinite series.
If a is a real number with absolute value |a| < 1, then the sequence an has limit 0. If 0 < a, then the sequence a1/n has limit 1.
Also:
The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence).
16. Representations The number e can be represented as a real number in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. The chief among these representations, particularly in introductory calculus courses is the limit
given above, as well as the series
given by evaluating the above power series for ex at x=1.
Still other less common representations are also available. For instance, e can be represented as an infinite simple continued fraction:
17. History of infinite sequences The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.
Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.
Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). In the latter work, Newton considers the binomial expansion of (x+t)n which he then linearizes by taking limits (letting t?0).
18. Formation of modern definition In the 18th century, mathematicians like Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his Thorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated under which conditions a series converged to a limit.
The modern definition of a limit (for any e there exists an index N so that ...) was given independently by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticed at the time) and by Cauchy in his Cours d'analyse (1821).
19. Comments The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence).
A sequence of real numbers may tend to infinity or minus infinity, compare infinite limits. Even though this can be written in the form
and
such a sequence is called divergent, unless we explicitly consider it a sequence in the affinely extended real number system or (in the first case only) the real projective line. In the latter cases the sequence has a limit (in the space itself), so could be called convergent, but when using this term here, care should be taken that this does not cause confusion.
20. A sequence is a function In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.
The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.
21. Limit of a function Suppose (x) is a real-valued function and c is a real number. The expression:
means that (x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of of x, as x approaches c, is L". Note that this statement can be true even if f(c)?L.
Indeed, the function (x) need not even be defined at c.
Consider f(x)=x/(x^2+1) as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:
f(1.9)= 0.4121
f(1.99)=0.4012
f(1.999)=0.4001
f(2)=0.4
f(2.001)=.3998
f(2.01)=.3988
f(2.1)=.3882
As x approaches 2, (x) approaches 0.4 and hence we have . In the case where , is said to be continuous at x = c, but it is not always the case.
22. Formal definition Whenever a point x is within d units of p, f(x) is within e units of L
Karl Weierstrass formally defined a limit as follows:
Let f be a real-valued function defined on an open interval of real numbers containing c (except possibly at c) and let L be a real number. Then
means that
for each real e > 0 there exists a real d > 0 such that for all x with 0<|x-c|<d, we have |f(x)-L|<e.
or, symbolically,
23. Limit of a function at infinity A related concept to limits as x approaches some finite number is the limit as x approaches positive or negative infinity. This does not literally mean that the difference between x and infinity becomes small, since infinity is not a real number; rather, it means that x either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity).
For example, consider
f(100) = 1.9900
f(1000) = 1.9990
f(10000) = 1.9999
For all x > S, f(x) is within e of L
As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, we say that the limit of f(x) as x approaches infinity is 2. In mathematical notation,
24. Limit of a sequence as a function Formally, we have the definition
if and only if for each e > 0 there exists an S such that
Note that the S in the definition will generally depend on e.
If one considers the domain of f to be the extended real number line, then the limit of a function at infinity can be considered as a special case of limit of a function at a point. Limit of a sequence is analogue of the limit of a function.
Formally, suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write
to mean
For every real number e > 0, there exists a natural number n0 such that for all n > n0, |xn-L| < e.
25. Properties Consider the following function: f(x)=xn if n-1<x=n. Then the limit of the sequence of xn is just the limit of f(x) at infinity.
A function f, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(xn)) converges to f(L) given that (xn) converges to L, i.e.
implies
Note that this equivalence does not hold in general for spaces which are not first-countable.
26. Convergence Compare the basic property (or definition):
f is continuous at x if and only if
A subsequence of the sequence (xn) is a sequence of the form (xa(n)) where the a(n) are natural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.
Every convergent sequence in a metric space is a Cauchy sequence and hence bounded. A bounded monotonic sequence of real numbers is necessarily convergent: this is sometimes called the fundamental theorem of analysis. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete.
A sequence of real numbers is convergent if and only if its limit superior and limit inferior coincide and are both finite.
27. Operations commuting with limit The algebraic operations are everywhere continuous (except for division around zero divisor); thus, given
and
then
and (if L2 and yn is non-zero)
These rules are also valid for infinite limits using the rules
q + 8 = 8 for q ? -8
q 8 = 8 if q > 0
q 8 = -8 if q < 0
q / 8 = 0 if q ? 8
28. Useful identities for limit of functions The following rules are valid if the limits on the right hand side exist (and are finite).
where S is a scalar multiplier.
where b is a positive real number.
if the right hand side makes sense; i.e., the two limits on the right exist, and
29. Cauchy sequence
The plot of a Cauchy sequence (xn), shown in blue, as n versus xn. If the space containing the sequence is complete, the "ultimate destination" of this sequence, that is, the limit, exists.
A sequence that is not Cauchy. The elements of the sequence fail to get close to each other as the sequence progresses.
30. Real numbers as Cauchy sequences A sequence
of real numbers is called Cauchy, if for every positive real number e, there is a positive integer N such that for all natural numbers m,n > N
where the vertical bars denote the absolute value.
In a similar way one can define Cauchy sequences of complex numbers. To define Cauchy sequences in any metric space, the absolute value | xm - xn | is replaced by the distance d(xm,xn) between xm and xn.
