Infinite series
Similar paradoxes occur in the manipulation of infinite series, such as 1/2 + 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 the partial 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 the infinite sum to be exactly 1. The illustrates this geometric series graphically by repeatedly bisecting a unit square. (Series whose successive terms differ by a common ratio, in this example by 1/2, are known as geometric series.)
Other infinite series are less well-behaved—for example, the series 1 − 1 + 1 − 1 + 1 − 1 + ⋯ . (2) If the terms are grouped one way, (1 − 1) + (1 − 1) + (1 − 1) +⋯, then the sum appears to be 0 + 0 + 0 +⋯ = 0. But if the terms are grouped differently, 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) +⋯, then the sum appears to be 1 + 0 + 0 + 0 +⋯ = 1. It would be foolish to conclude that 0 = 1. Instead, the conclusion is that infinite series do not always obey the traditional rules of algebra, 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 to converge, and the value 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 of calculus had an intuitive concept of limits, but it was only with the work of the German mathematician Karl Weierstrass that a completely satisfactory formal definition of the limit of a sequence was obtained.
Consider a sequence (an) of real numbers, by which is meant an infinite list a0, a1, a2, …. It is said that an converges to (or approaches) the limit a as n tends to infinity, if the following mathematical statement holds true: For every ε > 0, there exists a whole number N such that |an − a| < ε for all n > 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) approximates a within an error less than the chosen amount (|an − a| < ε). Stated less formally, when n becomes large enough, an can be made as close to a as desired.
For example, consider the sequence in which an = 1/(n + 1), that is, the sequence 1, 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 as n tends to infinity. The difference |an − 0| can be made smaller than any ε by choosing n sufficiently large. In fact, n > 1/ε suffices. 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 of continuity of a function. Intuitively, a function f(t) approaches a limit L as t approaches a value p if, whatever size error can be tolerated, f(t) differs from L by less than the tolerable error for all t sufficiently close to p. 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 function f(t) approaches a limit L as t approaches a value p 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 this context, it is straightforward to define continuity of a function. Continuous functions preserve limits; that is, a function f is continuous at a point p if the limit of f(t) as t approaches p is equal to f(p). And f is continuous if it is continuous at every p for which f(p) is defined. Intuitively, continuity means that small changes in t produce small changes in f(t)—there are no sudden jumps.
Properties of the real numbers
Earlier, the real numbers were described as infinite decimals, although such a description makes no logical sense without the formal concept of a limit. This is because an infinite decimal expansion such as 3.14159… (the value of the constant π) actually corresponds to the sum of an infinite series 3 + 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, the rational numbers) have important properties that correspond to intuitive notions of continuity. For example, consider the function x2 − 2. This function takes the value −1 when x = 1 and the value +2 when x = 2. Moreover, it varies continuously with x. It seems intuitively plausible that, if a continuous function is negative at one value of x (here at x = 1) and positive at another value of x (here at x = 2), then it must equal zero for some value of x that lies between these values (here for some value between 1 and 2). This expectation is correct if x is a real number: the expression is zero when x = Square root of√2 = 1.41421…. However, it is false if x is restricted to rational values because there is no rational number x for which x2 = 2. (The fact that Square root of√2 is irrational has been known since the time of the ancient Greeks. See Sidebar: 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 of completeness.
One awkward aspect of the concept of the limit of a sequence (an) is that it can sometimes be problematic to find what the limit a actually is. However, there is a closely related concept, attributable to the French mathematician Augustin-Louis Cauchy, in which the limit need not be specified. The intuitive idea is simple. Suppose that a sequence (an) converges to some unknown limit a. Given two sufficiently large values of n, say r and s, then both ar and as are very close to a, which in particular means that they are very close to each other. The sequence (an) is said to be a Cauchy sequence if it behaves in this manner. Specifically, (an) is Cauchy if, for every ε > 0, there exists some N such that, whenever r, 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 sequence converges. 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 numbers x, y, and z are:
- a. Trichotomy law. One and only one of the statements x < y, x = y, and x > y is true.
- b. Transitive law. If x < y and y < z, then x < z.
- c. If x < y, then x + z < y + z for all z.
- d. If x < y and z > 0, then xz < yz.
More subtly, the real number system is Archimedean. This means that, if x and y are real numbers and both x, y > 0, then x + x +⋯+ x > y for some finite sum of x’s. The Archimedean property indicates that the real numbers contain no infinitesimals. Arithmetic, completeness, ordering, and the Archimedean property completely characterize the real number system.