× will be less than any given number, or will differ α so little from 0 that its value may be disregarded without making any appreciable difference in the result; and the expression will then represent the true value of the sum 1 r of all the terms of the series. Whence we may conclude that the expression for the sum of the terms of a decreasing progression, in which the number of terms is infinite, is that is, equal to the first term, divided by 1 minus the ratio. That is, properly speaking, the limit to which the partial sums approach, as we take a greater number of terms in the progression. The difference between these sums and may be made as small as we please, but will only become inappreciable when the number of terms is infinite. a 1 -r We have, for the expression of the sum of the terms, The error committed by taking this expression for the value of the sum of the n first terms is expressed by The error committed by taking for the sum of a number of terms is evidently less in proportion as the number of terms is greater. 2. Again take the progression 213. In the several questions of geometrical progression. there are five numbers to be considered, — first, the first term, a; second, the ratio, r; third, the number of terms, n; fourth, the last term, 7; fifth, the sum of the terms, S. 214. We shall terminate this subject by solving this problem : Find a mean proportional between any two numbers, as m and n. Denote the required mean by x. and hence We shall then have (§ 197) Multiply the two numbers together, and extract the square root of the product. Exercises. 1. What is the geometrical mean between the numbers 2 and 8? Mean√8 x 2 = √16=4, Ans. 2. What is the mean between 4 and 16? 3. What is the mean between 3 and 27? 4. What is the mean between 2 and 72? 5. What is the mean between 4 and 64? Ans. 8. Ans. 9. Ans. 12. Ans. 16. CHAPTER X. LOGARITHMS. 215. The nature and properties of the logarithms in common use will be readily understood by considering attentively the different powers of the number 10. They are It is plain that the exponents 0, 1, 2, 3, 4, 5, etc., form an arithmetical progression of which the common difference is 1; and that the corresponding numbers 1, 10, 100, 1000, 10000, 100000, etc., form a geometrical progression of which the common ratio is 10. The number 10 is called the base of the system of logarithms; and the exponents 0, 1, 2, 3, 4, 5, etc., are the logarithms of the numbers which are produced by raising 10 to the powers denoted by those exponents. 216. If we denote the logarithm of any number by m, then the number itself will be the mth power of 10; that is, if we represent the corresponding number by M, The logarithm of a number is the exponent of the power to which it is necessary to raise the base of the system in order to produce the number. 217. If, as before, 10 denotes the base of the system of logarithms, m any exponent, and M the corresponding number, we shall then have 10m = M in which m is the logarithm of M. (1) If we take a second exponent, n, and let N denote the corresponding number, we shall have 10" N = in which n is the logarithm of N. (2) If, now, we multiply the first of these equations by the second, member by member, we have But, since 10 is the base of the system, m+n is the logarithm of M× N. Hence The sum of the logarithms of any two numbers is equal to the logarithm of their product. Therefore the addition of logarithms corresponds to the multiplication of their numbers. 218. If we divide Equation (1) by Equation (2), member by member, we have But, since 10 is the base of the system, m n is the loga |