the greatest (numerically). Then {(x+h)n−Sr}/h r+1 lies between n(r+1)xn−r−1 and n(r+1)xn−r−1(1+θ)n; and the difference between these can be made as small as we please by taking h small enough. Thus we can say that the limit of {(x+h)n−Sr}/hr+1 is n(r+1)xn−r−1; but the approach to this limit is of a different kind from that considered in (i.), and its investigation involves the idea of continuity.
V. Continuity.
59. The idea of continuity must in the first instance be introduced from the graphical point of view; arithmetical continuity being impossible without a considerable extension of the idea of number (§ 65). The idea is utilized in the elementary consideration of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.
60. The first step consists in the functional treatment of equations. Thus, to solve the equation ax2+bx+c = 0, we consider, not merely the value of x for which ax2+bx+c is 0, but the value of ax2+bx+c for every possible value of x. By graphical treatment we are able, not merely to see why the equation has usually two roots, and also to understand why there is in certain cases only one root (i.e. two equal roots) and in other cases no root, but also to see why there cannot be more than two roots.
Simultaneous equations in two unknowns x and y may be treated in the same way, except that each equation gives a functional relation between x and y. (“Indeterminate equations” belong properly to the theory of numbers.)
61. From treating an expression involving x as a. function of x which may change continuously when x changes continuously, we are led to regard two functions x and y as changing together, so that (subject to certain qualifications) to any succession of values of x or of y there corresponds a succession of values of y or of x; and thence, if (x, y) and (x+h, y+k) are pairs of corresponding values, we are led to consider the limit (§ 58 (ii.)) of the ratio k/h when h and k are made indefinitely small. Thus we arrive at the differential coefficient of ƒ(x) as the limit of the ratio of ƒ(x+θ)−ƒ(x) to θ when θ is made indefinitely small; and this gives an interpretation of nxn−1 as the derived function of xn (§ 45).
This conception of a limit enables us to deal with algebraical expressions which assume such forms as 00 for particular values of the variable (§ 39 (iii.)). We cannot, for instance, say that the fraction x2−1x−1 is arithmetically equal to x+1 when x = 1, as well as for other values of x; but we can say that the limit of the ratio of x2−1 to x−1 when x becomes indefinitely nearly equal to 1 is the same as the limit of x+1.
On the other hand, if ƒ(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which ƒ(y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.
62. The elementary idea of a differential coefficient is useful in reference to the logarithmic and exponential series. We know that log10N(1+θ) = log10N+log10(1+θ), and inspection of a table of logarithms shows that, when θ is small, log10(1+θ) is approximately equal to λθ, where λ is a certain constant, whose value is .434... If we took logarithms to base a, we should have
approximately. If therefore we choose a quantity e such that
which gives (by more accurate calculation)
The deduction of the expansions
is then more simply obtained by the differential calculus than by ordinary algebraic methods.
63. The theory of inequalities is closely connected with that of maxima and minima, and therefore seems to come properly under this head. The more simple properties, however, only require the use of elementary methods. Thus to show that the arithmetic mean of n positive numbers is greater than their geometric mean (i.e. than the nth root of their product) we show that if any two are unequal their product may be increased, without altering their sum, by making them equal, and that if all the numbers are equal their arithmetic mean is equal to their geometric mean.
VI. Special Developments.
64. One case of convergence of a sequence has already been considered in § 58 (i.). The successive terms of the sequence in that case were formed by successive additions of terms of a series; the series is then also said to converge to the limit which is the limit of the sequence.
Another example of a sequence is afforded by the successive convergents to a continued fraction of the form a0+1a1 + 1a2 +. . . , where a0, a1, a2, . . . are integers. Denoting these convergents by P0/Q0, P1/Q1, P2/Q2, . . . they may be regarded as obtained from a series P0Q0 + (P1Q1 − P0Q0 ) + (P2Q2 − P1Q1) + . . . ; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators 1 and denominators continually increasing.
Another kind of sequence is that which is formed by introducing the successive factors of a continued product; e.g. the successive factors on the right-hand side of Wallis’s theorem
A continued product of this kind can, by taking logarithms, be replaced by an infinite series.
In the particular case considered in § 58 (i.) we were able to examine the approach of the sequence S0, S1, S2, . . . to its limit X by direct examination of the value of X−Sr. In most cases this is not possible; and we have first to consider the convergence of the sequence or of the series which it represents, and then to determine its limit by indirect methods. This constitutes the general theory of convergence of series (see Series).
The word “sequence,” as defined in § 58 (i.), includes progressions such as the arithmetical and geometrical progressions, and, generally, the succession of terms of a series. It is usual, however, to confine it to those sequences (e.g. the sequence formed by taking successive sums of a series) which have to be considered in respect of their convergence or non-convergence.
In order that numerical results obtained by summing the first few terms of a series may be of any value, it is usually necessary that the series should converge to a limit; but there are exceptions to this rule. For instance, when n is large, n! is approximately equal to √(2πn)⋅(n/e)n; the approximation may be improved by Stirling's theorem
loge2 + loge3 + ... + loge(n−1) + 12logen = 12loge(2π) + nlogen − n + B11⋅2⋅n − B23⋅4⋅n3 + ... + (−)r−1Br(2r−1)⋅2r⋅n2r−1+ ...
,where B1, B2, . . . are Bernoulli's numbers (§ 46 (v.)), although the series is not convergent.
65. Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (§ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e. a number which cannot be expressed as the ratio of two integers.
These are isolated cases of irrational numbers. Other cases arise when we consider the continuity of a function. Suppose, for instance, that y = x2; then to every rational value of x there corresponds a rational value of y, but the converse does not hold. Thus there appear to be discontinuities in the values of y.
The difficulty is due to the fact that number is naturally not
