We conclude, therefore, that, if n be very large compared with a, the difference of the logarithms of n+x and n will be nearly proportional to the difference of the numbers n + x and x (Art. 869): and that if the difference of log (n + 1) and log n be given, the difference of log (n+x) and log n, when is small compared with n, may always be found approximately by a simple proportion. If the logarithms under consideration be tabular and not Napierian, then where is the modulus of tabular logarithms (Art. 907): if A therefore be assumed to be then the formula n log (n + x) = log n + n▲ will furnish an approximate value of log (n+x): it will be at once seen that this proposition is the basis of the rules in Art. 870, to which we have before referred. CHAPTER XXXVI. Limits of series. In a geometric ON THE LIMITS OF THE VALUES OF SERIES PROCEEDING AC- OR INCREASE. 915. THE following propositions relating to the limits of the values of series, proceeding according to ascending or descending powers of a symbol, which is capable of indefinite increase or diminution, will be found to be extremely useful in many enquiries, and more particularly in those which regard the application of Algebra to the properties of curvilinear figures. The complete theory of limits, properly so called, is only partially involved in them, and will be more expressly considered in a subsequent Chapter of this work. 916. If we make x = a + d' the sum of the geometric series series. is equal to a+d. a+ax+ax2 + ax3+... For the sum of this series, when x is less than 1, is a 1-x 1 It will follow, therefore, that a value of x, in such a series, may always be assigned, which will make its sum (s) differ from its first term a by a quantity less than any that can be assigned: for if be the quantity thus assigned, then any value of x which is less than a+ بره {say == 2(a+8)}, by a quantity less than d. مرده , will make s differ from a of a limit. 917. A limit of a series or expression is a fixed value to Definition which it approaches nearer than for any assignable difference, whilst a symbol upon which it is dependent is indefinitely diminished or indefinitely increased, but which it never attains whilst the value of that symbol is different from zero in one case or from infinity in the other. 918. It will follow therefore that the limit of the value of The limit the series a+ax+ax2 + ax3+ ... of a geo metric series proceeding acx cording to the ascend a ing powers is its first term or a: for it has been shewn that a value of may be assigned, which will make its sum differ from a by quantity less than any other that can be assigned: and it never of a symbol attains to this limit, though it may approach indefinitely near term. to it, whilst x is different from zero. 919. If in the series is its first The limit of a geometric series pro proceeding according to inverse powers of x, we make x= a+d ceeding according δ then its sum is equal to a + d. to inverse powers of a symbol. It will follow, therefore, that, in such a series, a value of ≈ may always be assigned, which will make its sum (s) differ from its first term by a quantity less than any which may be assigned: for if d be the quantity thus assigned, then any value a+d of a greater than {say 2 (a+b)} will make s less than a + d. The limit, therefore, of the value of this series, when x is indefinitely (see Chap. xxxvII.) great, is its first term or a. 920. The terms of the geometric series a+arx+ar2x2+ ar2x2+ Formation of a geo metric are severally either equal to or greater than the corresponding is a supeterms of the series ceeding ac- if r be the greatest inverse ratio of any two consecutive coefficients. cording to powers of the same symbol. In the first place, let be greater than 2, 3, and all sub a1 sequent ratios of a similar kind: then we have a=r, and a therefore a1 = ar : is less than r, and therefore a, is less Ag a than air, and therefore also less than ar3, since a1 = ar: is less than r, and therefore a, is less than a,r, and therefore less than ar, since a, is less than ar2: and similarly for all subsequent coefficients of the series: it follows, therefore, that the first and second terms of the geometric series (a) are equal to the first and second terms of the series (3), but that all the subsequent terms of the first series are severally greater than those corresponding to them in the second. In the second place, let any other inverse ratio, such as than ar: a less than r, and therefore a, less a less than r, and therefore a, less than a、r, and a fortiori less than ar2, since a, is less than ar: and so on, until we come to a-1, which is less than ar: the next ratio ап an-1 = =r, and therefore a, a,-ir, which is less than ar", since a, is less than ar-1: and similarly for all subsequent coefficients: it will follow, therefore, that the terms of the series (a), after the first, are severally greater than those corresponding to them in the series (ß). an an-1 be equal to each other, but greater than all those which follow them, then if =r, then first terms of the series, (a) a and (6) are equal to each other: but all the subsequent terms of the series (a) are severally greater than those corresponding to them in the series (B). We may assume the coefficients of the series (6) to be positive, and to increase perpetually as we recede from the first term: but the proposition which we have demonstrated above will be true a fortiori if the coefficients form a decreasing series, or if one or more of them become zero or negative. It will follow generally, therefore, that if r be the greatest inverse ratio of any two consecutive coefficients, the terms of the geometric series (a) are severally equal to or greater than the corresponding terms of the series (B), and that consequently a value of may always be determined, which will make the sum of the series (3) differ from its first term by a quantity 8 less than any which may be assigned: thus, if ræ= and a + 8' d therefore - the sum of the series (a) is a + d, and r(a+d)' therefore the sum of the series (6) is less than a +d: and the limit of its value (Art. 917) is a, or the first term of the series. Thus, in the series 1 × 2 + 2 × 3x + 3 × 4 x2 + 4 × 5 x3 + ... Examples. (B), the value of r or of the greatest inverse ratio of two consecutive coefficients is 3: and the terms of the geometric series are severally equal to or greater than those of the series pro posed (6): if x = 8 the sum of the terms of the series (B), will differ from its first term 2 by a quantity less than d. If the series proposed had been 1 + 1 × 2x + 1 × 2 × 3x2 + 1 × 2 × 3 × 4x3 + the ratio of the nth to the (n − 1)th coefficient is n, which increases indefinitely the series is therefore infinite, if indefinitely continued, whatever be the value of x. : limit of a term is 921. PROPOSITION. If there be three quantities whose On the values are expressed by series proceeding according to powers series of the same symbol, and if, for the same value of that symbol, whose first the first be necessarily greater than the second and the second unknown, than the third; then, if the first and third series have the same is included first term or the same limit, the first term or limit of the second in value beVOL. II. NN but which |