Infinite Series

CHAPTER I

FUNCTIONS AND LIMITS

1. Introduction. The theory of infinite series is an important branch of elementary mathematical analysis. For its proper understanding it is essential for the reader to have some knowledge of such fundamental ideas as bounds, limits, continuity, derivatives and integrals of functions. In this chapter a brief sketch will be given of those results in analysis which will be used in the book, and also a more detailed discussion of the question of limits. It will be assumed that the reader is familiar with the simple properties of the logarithmic, exponential, hyperbolic and circular functions. Certain properties of these functions, however, which are of special importance in the theory of series will be derived in Art. 18.

2. Functions. It is sufficient for our purpose to regard a function of a variable as a mathematical expression which possesses one calculable value corresponding to each of a set of values of the variable. Each calculated value of the expression is called the value of the function corresponding to the appropriate value of the variable. Throughout, the letter x or y will denote a real variable, that is, a variable which takes only real values and, unless otherwise stated, the functions with which we deal will also be assumed to be real, that is, to possess only real values. Functions of x are usually denoted by symbols such as F(x), f(x), φ(x), etc., and their values when x = a by F(a), f(a), φ(a), etc. If values of the function f(x) can be determined for certain values of the variable x we say that f(x) is defined for these values of x. If the function f(x) is defined for all values of x satisfying the inequality awe say that f(x) is defined in the open interval (a, b). If, in addition, f(x) is defined for x = a and for x = b, then f(x) is defined for a≤x≤b and we say that f(x) is defined in the closed interval (a, b).

3. Bounds of a Function. Suppose that the function f(x) is defined for a certain set of values of x. If there is a number which is greater than all the values of f(x) then f(x) is said to be bounded above for these values of x. If there is a number which is smaller than all the values of f(x) then f(x) is said to be bounded below for these values of x. If both conditions are satisfied f(x) is said to be bounded for these values of x.

If, for a certain set of values of x, there is a number K, independent of x, such that (i) f(x) ≤ K, (ii) there is at least one value of x for which f(x) > K − ε, where ε is any positive number,¹ then K is called the upper bound of f(x) for this set of values of x. If, for a certain set of values of x, there is a number k, independent of x, such that (i) f(x) ≥ k, (ii) there is at least one value of x for which f(x) < k + ε, then k is called the lower bound of f(x) for this set of values of x.

It is clear that the lower bound of f(x) is not greater than its upper bound.

The functions

are, respectively, unbounded above and bounded below with lower bound zero, unbounded above and below, bounded above and below with upper and lower bounds +1 and −1.

The following theorem is fundamental.²

THEOREM A. If f(x) is bounded above for a certain set of values of x, it possesses an upper bound for these values of x. If f(x) is bounded below it possesses a lower bound.

4. Limits of Functions. The function f(x) is said to tend to the limit l as x tends to a if, given ε, we can ³ find η = η)(ε) such ⁴ that |f(x)−l| < ε for all values of x except a for which the function is defined and which also satisfy the inequality |x−a| < η. In these circumstances we write e9780486154855_i0003.jpg

The function f(x) is said to tend to the limit l as x tends to infinity if, given ε, we can find X = X(ε) such that |f(x)−l∣ < ε for all values of x > X for which the function is defined. In these circumstances we write e9780486154855_i0004.jpg

The function f(x) is said to tend to infinity as x tends to infinity if, given any positive number K, we can find X = X(K) such that f(x) > K for all values of x > X for which the function is defined. In these circumstances we write f(x)→∞ as x→∞ or e9780486154855_i0005.jpg

The reader should also construct definitions corresponding to the expressions

Throughout the remainder of this article we shall consider only limits as x→∞ and we shall assume that the functions under discussion are defined for all sufficiently large values of x. There being no possibility of ambiguity we shall use the contracted notation lim f(x) for a limit of this kind. The subsequent theorems hold with trivial modifications for other types of limits and, in particular, for limits as x→∞ through a certain set of values. It will be observed that from the definition of a limit it follows that, if f(x) is defined for all values of x and if lim f(x) = l, then a fortiori f(x)→l as x tends to infinity through any set of values and, in particular, through all positive integral values. We now prove some fundamental theorems on limits.

THEOREM 1. If lim f1(x) = l1, lim f2(x) = l2, then

(i) lim {f1(x) + f2(x)} = l1 + l2,

(ii) lim f1(x) f2(x) = l1l2,

(iii) lim f1(x)/f2(x) = l1l2,

where, in (iii), l2 ≠ 0.

Corresponding to any positive number θ we can find X1 = X1(θ), X2 = X2(θ) such that

|f1(x)−l1| < θ, |f2(x)−l2| < θ,

whenever x > X1, x > X2 respectively. If X = Max (X1, X2), that is, if X is the larger of X1 and X2, then these two inequalities hold a fortiori whenever x > X.

(i) Given ε, let e9780486154855_i0007.jpg and determine X as above. Then whenever x > X, which depends only on ε,

and this proves (i).

(ii) Given ε, let θ be the positive root of the equation

x² + (|l1|+|l2|)x − ε = 0,

and determine X as a function of θ, and therefore of ε, as before. Then, whenever x > X,

which proves (ii).

(iii) Given ε, let θ be any positive number satisfying both the inequalities

and determine X as before. Then,⁵ whenever x > X,

which proves (iii).

It should be noted that (i) and (ii) hold not merely for two but for any finite number of functions f1(x), f2(x), f3(x), ... The reader should examine how far the theorem remains true in the case when either l1 or l2 or both are infinite.

THEOREM 2. If, for all sufficiently large values of x, we have f1(x) ≤ f2(x) then l1 ≤ l2, where l1 = lim f1(x) and l2 = lim f2(x) and it is assumed that these limits exist.

Suppose if possible that l1 > l2. Let ∊ be e9780486154855_i0012.jpg Then, as in the proof of Theorem 1, we can determine X = X(ε) such that, whenever x > X,

l1 −ε <f1(x)<l1 + ε, l2 − ε<f2(x)<l2 + ε.

It follows that, for such values of x,

f1(x) − f2(x) > l1 − ε − l2 − ε = 0,

which contradicts the hypothesis.

The theorem is therefore proved.

COROLLARY. Under the conditions of Theorem 2 if lim f1(x)= ∞, then lim f2(x) = ∞.

It should be noted in passing that the hypothesis f1(x) < f2(x) does not imply the conclusion l1 < l2. For