Limit of Sequence
is a function defined on a set of real numbers for all positive integers.
The intuitive definition of the sequence is already outlined in the Introduction. Roughly speaking,
Definition 1:has a limit L if (xn-L) becomes arbitrary small for all sufficiently large values of n. In this case we write
(2) .
From this crude description, we would expect that the sequence 1,1,… has the limit 1, whereas the
sequence has a limit 0 while the sequence 1,-2,3,-4,… does not have a limit. On the other hand, our intuition is not sharp enough to deduct, whether the sequence has a limit and to compute the limit if there is one. Even a relatively simple sequence as , -1,1, -1,…, so-called oscillated sequence, could lead to the wrong intuitive conjecture, that it has two limits. So, we need an accurate definition of the ‘limit of a sequence’ based on which we would become capable to predict for any given sequences the existence of its limit and to evaluate it.
We emphasise that limit L should be a real number. Formally, Definition 1 means, that
Definition 2: (3) .
We can interpret it as follows: the proof that L is the limit of a given sequence , consists upon being given an of finding a value of N, such that the inequality holds for all values of n except at most a finite number, namely . The value of N will, in general, depend on the value of . Figure 1 illustrates Definition 2. All of the xn, except at most a finite number of terms, must be inside the parentheses.
x1 XN-1 xN+1 xN+2 XN x2 x3 L - L L+ Figure 1
How Definition 2 works in practice?
Example 1: Consider the sequence , which can be expressed as xn=1/n (n=1,2,3,…). We already made a conjuncture that . Let us prove it.
Following Definition 2 for given we must find N so that for all n>N: .
Substitution of xn=1/n into the last inequality leads to
, or, considering n>0 (4) .
Thus if we choose N so, that 1/N<, then certainly (4) will hold since for all n larger than N. Now iff . Hence if we take any integer N such that , then (3) will hold for the given sequence with L=0. This proves thatalthough not one term of the sequence is equal to zero.
Example 2 Examining the sequence xn=1(n=1,2,3,…) in terms of Definition 2 we can prove that our guess was correct. Really, if L=1 then for any . So in this case (3) always holds and N does not depend on.
Example 3. Consider the sequence 1,2,3,…, which can be expressed as .It can be proved by contradiction using the same -N method, which leads to the statement that the sequence 1,2,3,… has no limit or, that tends to infinity when n tends to infinity or diverges to infinity.
Infinity is certainly not a number. Moreover, it is not a quantative concept. It is a quality of increase beyond bound. Although the concept of infinity is difficult to grasp we can define it as not finite, contrary to finite, which is completely determinable by counting or measurement. Following the Galileo statement that there are as many squares as there are natural numbers, G. Cantor proved that the set of all integers, the set of all natural numbers, the set of all rational numbers and the set of all algebraic numbers are equivalent to the set of all natural numbers as they can be put in one-to-one correspondence with the infinitude of natural numbers. Following this concept we may think of all divergent to infinity sequences as having the ‘same manyness of elements’ as, by the definition, each sequence has one- to one correspondence with the set of all positive integers, hence belong to ‘aleph null’ set of infinity (1p.258-264). Remembering though, that infinity cannot be expressed by any number (other things, like motherhood, happiness, faith belong to the qualitative category of concepts, that humans were hopelessly trying to describe by quantity), we discourage the idea to resolve indeterminate problems of substituting each term of the former by equal numbers in case of; for example, {} simply because there are no numbers equal to infinity.
Coming back to Example 3 whatever large number we choose there are always terms, which would exceed it. This reasoning seems to breach Definition 2. Really, we determine infinite as not a number, hence in this case the inequality makes no sense. So Definition 2 needs amendments such that there would not be a need to use the ‘suspicious’ concept in notation. We have already proved, that a sequence may have a limit, which is different to any of its terms, so the fact that ‘’is not a number should not contradict the perception of a limit. Following experience with the sequence=n we know, that
Definition 3 For any given positive number M, there is an integer N (depending on M) such, that xn>M for every .
This definition binds concepts of convergence to a finite limit as to a real number and divergence to infinity.
. Is a sequence whose terms get ‘ too big’ as in Example 3, the only kind, which does not have a limit? We already consider a ‘suspicious’ example Let us suppose that it has a limit, so . Definition 2 has to be satisfied for any . Let us choose . Following (3) there would be a positive integer N such as .
If n is even then the last expression means:/1-L/< ½ and for n odd it is /-1-L/<1. This implies that L should be less than half unit from 1 and less that half unit from –1, which is impossible. So by contradiction we proved that the sequence 1,-1,1,-1,… has no limit though the terms of the sequence all have absolute value 1 and hence are not ‘too big’. It is worth noticing, that the initial guess, that there may be two limits of the above sequence is proved to be wrong.
The last example illustrates one very important property of limit of sequence: a sequence has at most
one limit.
Limit of Series
Another very important application of the concept under discussion is the limit of series of real numbers, which already appeared in the Zeno Paradox (see formula (1)). Formally
Definition 4 Infinite series of real numbers can be defined as an ordered pair , where is a sequence and is called the nth partial sum of the series that forms a new sequence {sn}, which can either converge to the limit S, if the limit does exist , or diverge, if does not exist or if sn diverges to .
From the above definition it is easy to deduce, that the behaviour of series depends on the behaviour of the sequence of its partial sums. Moreover, we can make a conjuncture, that for partial sums to converge to a real number, the limit of the term an ought to become ‘smaller and smaller’, or, formally, tend to zero as n tends to infinity. Unfortunately, this property is not sufficient to ensure, that be convergent. On the other hand, it can be used to determine a divergence of a series if .
Following example illustrates the point of the discussion.
Consider series . From Chapter 1 we know that . However, it can be proved, that sn diverges when . This is an example of a harmonic series, which terms are reciprocal to arithmetic sequences. In music, vibrating strings of the same material and with equal diameter, equal torsion and equal tension and whose lengths are proportional to terms in a harmonic sequence generate harmonic tones. Referring to A. Pushkin his personage Salieri may not be too wrong trying to test ‘ harmonies by algebra’. Knowing much more about harmonic series than people in XVIII century, we would rather opt for the real
analyses than for algebra.
.
Limit of Function of the Real Variable
Chapters 1 and 2 were concerned in mapping discrete integers in set of real numbers. We are now interested in mapping an interval A of real numbers in the set of real numbers B.
Definition 4 We say that f is a function from A to B if for every there is exists a unique such that , or
By contrast to a rational number, which caused concern to the Greek mathematicians due to the lack of completeness, real numbers R possesses the completeness property. This implies that real numbers in an interval A cannot be ordered by corresponding integer as we do for sequences. We order rational numbers by comparison of their values. So the definition we used for the limit of sequences and series needs adoption with respect to the completeness property of the former. Let and let f be a real-valued function whose domain includes all points in some open, interval (a-h, a+h) except, possibly, the point a itself.
Definition 5 We say that f(x) approaches L (where as x approaches a if given, there exists such thatas .It is worth noting that point a need not be in the domain of f. Thus well known although the function is not defined in n=0.
Definition 5 can be graphically illustrated (see Figure 2).
a- a a+
L- L- Figure 2
In order for f(x) to approach L as x approaches a the following must be true: given any parentheses about L there must existparentheses about a such that every arrow which begins inside the parentheses (except, possibly the arrow if there is one, that starts at a) must end inside the parentheses.
Roughly speaking, the following can be seen on the graph of a function f such that (Figure 3): as the x coordinate of a moving point of the graph gets close to a (from either the right or the left), the height f(x) of the point heads toward L. Thus both lines introduce functions y=f(x) (in red) and y=g(x) (in green) satisfy . Moreover, following Definition 5 even if f(x) and g(x) are defined such that f(2) and g(2) are not equal to 3, both functions still have 3 as their limit in x=2.
Figure 3 Figure 4
y y=f(x) y y=h(x)
L= 3 L= 3 L= 2
0 1 a= 2 x 0 1 a= 2 x
On the other hand, the function y=h(x) in Figure 4 has no limit at a=2 because h(x) gets close to 3 when
x gets close to a in the left, while h(x) gets close to 2 when x gets close to a on the right. From the uniqueness property of a limit hence there is no single number L such that h(x) gets close to L when x gets close to a, we
deduct that h(2) has no limit.
Finally, let us illustrate function (). Here (see Figure 5) as x gets close to a=0, the value f(x) oscillates rapidly. Even if we look only at one side of a, it is clear that there is no number L toward which the value f(x) tends. Hence f has no limit at 0.
To emphasise the strong analogy between Definitions 4-5 let us fill the ‘Table of analogues’ (see p.9). If each entry in the right-hand column is substituted for the corresponding entry on the left, we change Definition 4 into Definition 5. However, more than mechanical process is involved here. Corresponding entries in the table actually have the same meaning. For example is a function of an integer variable.
Furthermore, all algebra operations hold on both concepts with exceptions when we have indeterminate limits, such as. Many limits of an indeterminate form can be evaluated by a L’Hospitale formula, which uses methods of differential calculus.
Contrary, the forms are evidently determinate, in the sense that, for instance, if then
Other determinate forms are .
Expressions of the form a/0, where a is a non-zero number or are undefined, because if y is very small, then a/y will be very large in size but positive or negative according to the sign of y and a. Also is undefined, because ,but if g(x) equals the greatest odd integer .
. Table of analogues
Limit of series n sn L N
Limit of function (RV) f(x) x f(x) L a
As has been stated one of the most important application of the concept of limit is in conjunction with concept of continuity. Intuitively we can deduce that continuous function at a point has no gaps in this point. Formally, using limit concept, we can express this as
Definition 6 We say that the function f of a real variable is continuous at a point a if .
Definition 6 really demands that 2 conditions be fulfilled in order that f be continuous at a. The first condition is that the must exist; the second is that this limit be equal to f(a). In particular, if f(x) is not defined, thenf cannot be continuous at a. For example, the function f=sinx/x (x=0) is not defined at x=0 and hence is not continuous at x=0 even though its limit exists and equal to 1. we can define f(0)=1, then it becomes continuous at (0) since .For the fact that continuity is defined using limit we can deduct that all algebra operations hold for continuous functions.
Conclusions
People use words ‘infinity’, ‘limit’, ‘continuity’ every day: ‘government applies a limit to someone’s activity’, ‘someone limits one’s ambitions or aspirations’; ‘one’s waiting time on the NHS lasts for an infinite period’, ‘someone may express love as being infinite’; ‘one cannot complete refurbishment because the production of the wallpaper he/she needs is discontinued’, etc.
We face those concepts everywhere. Musical harmonies have pure mathematical structures and obey rules of harmonic series, which have an explicit relation with limit. The complexity of the concept made it one of the most important in philosophy. Theology appeals to it in the most important doctrines. On the other hand, scientists use these common words to define fundamental concepts of mathematical knowledge. The concept of limit led to differential and integral calculus and modern methods of approximation, which has an infinite variety of applications in modern physics, astronomy, chemistry, engineering, and biology. However, as we emphasised throughout the essay the concept of limit is still beyond understanding in a number of cases; the use of the Cauchy definition solved old problems, there exist undefined or indeterminate forms of limit, that are still unsolved.
Just a final remark: re-phrasing Ludwig Witgenstein ‘ the limits of my words mean the limits of my world’ (3, p.826). We strongly believe that although we are still limited by our English, the presented discussion delivers and clarifies our view on the most important and conceivably the most difficult concept in mathematics.
Appendix 1
Achilles and the tortoise
We provide calculations supporting the Paradox based on the assumptions:
- tortoise speed- 3m/sec (constant)
- Achilles speed- 6 m/sec (constant)
- head start given to tortoise- 3 meters
- first step:
distance run by Achilles: 3m time spent:
by tortoise: 3m/sec x 0.5 sec=1.5 m
- second step:
distance run by Achilles:1.5m time spent:;
by tortoise:3m/sec x 0.25sec=0.75 m
distance run by Achilles:0.75m time spent:;
by tortoise:3m/sec x 0.125sec=0.375 m, etc.
Summing up all these infinite number of interval leads to the infinite geometric sequence with the first term a=½ and the ratio r=½. The definition of the concept of limit following development of the real analysis resulted in the formula of convergence of the geometric series with –1< r <1, which turned the sum of infinite number of terms into a finite figure. Thus,
.
References and bibliography
1 Gullberg, J., Mathematics From the Birth of Number, 1997,W.W Norton& Co
2 Clark, C., Elementary Mathematical Analysis, Second Edition, Cobb& Dunlop Publisher Services
3 Knowles, E., The Oxford Dictionary of Quotations, 1999, Oxford University Press
4 Goldberg, R., Methods of Real Analyses’ Second Edition, John Willey& Sons, Inc, New York
5
* Illustrations composed of ‘Clip Art-on-line’ resources.
