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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 .

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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. 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. 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.

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 http://mathworld.wolfram.com/CardinalNumber.html 