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Similarparadoxes occur in the manipulation ofinfinite series, such as1/2 +1/4 +1/8 +⋯ (1) continuing forever. This particular series is relatively harmless, and its value is precisely 1. To see why this should be so, consider thepartial sums formed by stopping after a finite number of terms. The more terms, the closer the partial sum is to 1. It can be made as close to 1 as desired by including enough terms. Moreover, 1 is the only number for which the above statements are true. It therefore makes sense to define theinfinite sum to be exactly 1. Thefigure illustrates thisgeometric series graphically by repeatedly bisecting a unit square. (Series whose successive terms differ by a commonratio, in this example by1/2, are known as geometric series.)
Other infinite series are less well-behaved—for example, the series1 − 1 + 1 − 1 + 1 − 1 + ⋯ . (2) If the terms are grouped one way,(1 − 1) + (1 − 1) + (1 − 1) +⋯, then the sum appears to be0 + 0 + 0 +⋯ = 0. But if the terms are grouped differently,1 + (−1 + 1) + (−1 + 1) + (−1 + 1) +⋯, then the sum appears to be1 + 0 + 0 + 0 +⋯ = 1. It would be foolish to conclude that 0 = 1. Instead, theconclusion is that infinite series do not always obey the traditional rules ofalgebra, such as those that permit the arbitrary regrouping of terms.
The difference between series (1) and (2) is clear from their partial sums. The partial sums of (1) get closer and closer to a single fixed value—namely, 1. The partial sums of (2) alternate between 0 and 1, so that the series never settles down. A series that does settle down to some definite value, as more and more terms are added, is said toconverge, and thevalue to which it converges is known as the limit of the partial sums; all other series are said to diverge.
The limit of a sequence
All the great mathematicians who contributed to the development ofcalculus had an intuitive concept oflimits, but it was only with the work of the German mathematicianKarl Weierstrass that a completely satisfactory formal definition of the limit of asequence was obtained.
Consider a sequence (an) of real numbers, by which is meant an infinite lista0,a1,a2, …. It is said thatan converges to (or approaches) the limita asn tends toinfinity, if the following mathematical statement holds true: For every ε > 0, there exists awhole numberN such that |an −a| < ε for alln >N. Intuitively, this statement says that, for any chosen degree of approximation (ε), there is some point in the sequence (N) such that, from that point onward (n >N), every number in the sequence (an) approximatesa within anerror less than the chosen amount (|an −a| < ε). Stated less formally, whenn becomes large enough,an can be made as close toa as desired.
For example, consider the sequence in whichan = 1/(n + 1), that is, the sequence1,1/2,1/3,1/4,1/5, …, going on forever. Every number in the sequence is greater than zero, but, the farther along the sequence goes, the closer the numbers get to zero. For example, all terms from the 10th onward are less than or equal to 0.1, all terms from the 100th onward are less than or equal to 0.01, and so on. Terms smaller than 0.000000001, for instance, are found from the 1,000,000,000th term onward. In Weierstrass’s terminology, this sequence converges to its limit 0 asn tends to infinity. The difference |an − 0| can be made smaller than any ε by choosingn sufficiently large. In fact,n >1/εsuffices. So, in Weierstrass’s formal definition,N is taken to be the smallest integer >1/ε.
This example brings out several key features of Weierstrass’s idea. First, it does not involve any mystical notion of infinitesimals; all quantities involved are ordinary real numbers. Second, it is precise; if a sequence possesses a limit, then there is exactly one real number that satisfies the Weierstrass definition. Finally, although the numbers in the sequence tend to the limit 0, they need not actually reach that value.
Continuity of functions
The same basic approach makes it possible to formalize the notion ofcontinuity of a function. Intuitively, afunctionf(t) approaches a limitL ast approaches a valuep if, whatever size error can be tolerated,f(t) differs fromL by less than the tolerable error for allt sufficiently close top. But what exactly is meant by phrases such as “error,” “prepared to tolerate,” and “sufficiently close”?
Just as for limits of sequences, the formalization of these ideas is achieved by assigning symbols to “tolerable error” (ε) and to “sufficiently close” (δ). Then the definition becomes: A functionf(t) approaches a limitL ast approaches a valuep if for all ε > 0 there exists δ > 0 such that |f(t) −L| < ε whenever |t −p| < δ. (Note carefully that first the size of the tolerable error must be decided upon; only then can it be determined what it means to be “sufficiently close.”)
Having defined the notion of limit in thiscontext, it is straightforward to definecontinuity of a function. Continuous functions preserve limits; that is, a functionf is continuous at a pointp if the limit off(t) ast approachesp is equal tof(p). Andf is continuous if it is continuous at everyp for whichf(p) is defined. Intuitively, continuity means that small changes int produce small changes inf(t)—there are no sudden jumps.
Properties of the real numbers
Earlier, thereal numbers were described as infinite decimals, although such a description makes no logical sense without the formalconcept of a limit. This is because an infinite decimal expansion such as 3.14159… (the value of theconstant π) actually corresponds to the sum of an infinite series3 +1/10 +4/100 +1/1,000 +5/10,000 +9/100,000 +⋯, and the concept of limit is required to give such a sum meaning.
It turns out that the real numbers (unlike, say, therational numbers) have important properties that correspond to intuitive notions of continuity. For example, consider the functionx2 − 2. This function takes the value −1 whenx = 1 and the value +2 whenx = 2. Moreover, it varies continuously withx. It seems intuitively plausible that, if acontinuous function is negative at one value ofx (here atx = 1) and positive at another value ofx (here atx = 2), then it must equal zero for some value ofx that lies between these values (here for some value between 1 and 2). This expectation is correct ifx is a real number: the expression is zero whenx =Square root of√2 = 1.41421…. However, it is false ifx is restricted to rational values because there is no rational numberx for whichx2 = 2. (The fact thatSquare root of√2 isirrational has been known since the time of the ancient Greeks.SeeSidebar: Incommensurables.)
In effect, there are gaps in the system of rational numbers. By exploiting those gaps, continuously varying quantities can change sign without passing through zero. The real numbers fill in the gaps by providing additional numbers that are the limits of sequences of approximating rational numbers. Formally, this feature of the real numbers is captured by the concept ofcompleteness.
One awkward aspect of the concept of the limit of a sequence (an) is that it can sometimes be problematic to find what the limita actually is. However, there is a closely related concept, attributable to the French mathematicianAugustin-Louis Cauchy, in which the limit need not be specified. Theintuitive idea is simple. Suppose that a sequence (an) converges to some unknown limita. Given two sufficiently large values ofn, sayr ands, then bothar andas are very close toa, which in particular means that they are very close to each other. The sequence (an) is said to be aCauchy sequence if it behaves in this manner. Specifically, (an) is Cauchy if, for every ε > 0, there exists someN such that, wheneverr,s >N, |ar −as| < ε. Convergent sequences are always Cauchy, but is every Cauchy sequence convergent? The answer is yes for sequences of real numbers but no for sequences of rational numbers (in the sense that they may not have a rational limit).
A number system is said to be complete if every Cauchy sequenceconverges. The real numbers are complete; the rational numbers are not. Completeness is one of the key features of the real number system, and it is a major reason why analysis is often carried out within that system.
The real numbers have several other features that are important for analysis. They satisfy various ordering properties associated with the relation less than (<). The simplest of these properties for real numbersx,y, andz are:
- a. Trichotomy law. One and only one of the statementsx <y,x =y, andx >y is true.
- b. Transitive law. Ifx <y andy <z, thenx <z.
- c. Ifx <y, thenx +z <y +z for allz.
- d. Ifx <y andz > 0, thenxz <yz.
More subtly, the real number system is Archimedean. This means that, ifx andy are real numbers and bothx,y > 0, thenx +x +⋯+x >y for somefinite sum ofx’s. The Archimedean property indicates that the real numbers contain no infinitesimals. Arithmetic, completeness, ordering, and the Archimedean property completely characterize thereal number system.