The terms in the quotient are found thus; dividing the first remainder, x2, by a, the first term of x2 the divisor a-x, we shall have for the second α term of the quotient, because the division can only α be indicated; multiplying the divisor by and subtracting the product from x2, the remainder is X3 a ; again, dividing this remainder by a, the result will be 3 which is the third term in the quotient; and, in like manner, we might continue the operation as far as we please: But the law of continuation is evident, because the powers of x increase by unity in each successive term of the quotient, and the powers of a increase by unity in the denominator of each of the terms after the first, And the sum of the terms infinitely continued is said to be equal to the original fraction ax Thus we say that the numerical fraction, when reduced to a decimal, is equal to .6666, &c., continued to infinity. Ex. 2. It is required to convert α a X into an In this example, if x be less than a, the series is convergent, or the value of the terms continually diminishes; but, when x is greater than a, it is said to diverge: Thus, let a=3 and x=2, then 14 3C 302 x03 3 -+ ++, &c. =1+3+1+27+, &c.; where a a2 a3 the fractions or terms of the series grow less and less, and the farther they are extended the more they converge or approximate to 0, which is supposed to be the last term or limit. 30 x2 X3 But if a 2, and x=3, then 1++ ++, &c. α a2 ૮૨ =1+3+2+7+, &c., in which the terms become larger and larger. This is called a diverging series. 1 Ex. 3. It is required to convert into an infinite series. 1+a 178. If we make a=1, we have this remarkable comparison: 1 1+a =1-1+1-1+1-1+1-,&c. to infinity; which appears rather contradictory; for, if we stop at 1, the series gives 0; and if we finish at +1, it gives +1. The real question, however, results from the fractional parts, which (by division) is always + when the sum of the terms is 0, and when the sum is +1: because the complete quotient is found by placing the remainder over the divisor, in the form of a fraction, and annexing it to the terms in the quotient with its proper sign; but the remainder in the present case is +1, or -1; hence the fraction to be added is +, or -1; and, consequently, is the true quotient in the former case, and 1-, or in the other. This will appear evident by taking successive portions of the series: thus, for six terms, we shall have 1-1+1-1+1 -1+1=1, and for seven terms, 1-1+1-1+1 -1+1=1. SCHOLIUM. Here we might infer, by conversion, that the sum of an infinite series is found, when we know the fraction which would produce such a series by actual division; but, although it is a fact that the fraction is a value of the series, still it may not be the only one which would produce the same series: Thus, the above series, 1-1+1-1+1−1 +1-1, &c., to infinity, can be produced by several other fractions besides the fraction. Let, for example, be converted into an infinite ૐ series by actual division: Now, it is plain that = 1 1+1+12 and the operation will stand thus: -1−1 | 1−1+1−1+1−1+, &e.. +1 1 -1-1-1 +1, &c. In like manner, will produce the above series, and so on. 179. Let us now make a=1, and the preceding developement shall be ==1-+-+-+, &c.: The sum of two terms is, which is too small by ; three terms give, which is too much by for the sum of four terms, we have, which is too small by, &c. We see here that the successive portions of the series are alternately greater and less than the fraction, which represent it; but that the difference, whether it be in excess or deficiency, becomes less and less. 1 180. Suppose again a=1, and we shall have 1+ Now, by considering only two terms, we have 2 which is too small by; three terms make 3, which |
