CHAPTER XXXI. INDETERMINATE COEFFICIENTS. 355. Convergent and Divergent Series. If the sum of the terms of a series approaches a fixed finite value when the number of terms is indefinitely increased, the series is Convergent. 356. If the sum of the terms of a series increases without end, or oscillates in value without approaching a fixed finite value when the number of terms is indefinitely increased, the series is Divergent. 357. The theorem for expanding expressions into series is called the Theorem of Indeterminate Coefficients, and is as follows: 358. If the series A+ Bx + Сx2 + Dx3 + ...is always equal to the series A' + B'x + C'x2 + D'x3 + ··, when x has values which make both convergent, the coefficient of the like powers of x will be equal. That is, A A', B = B', etc. Now if the equation = = A — A' — (B' — B)x + (C' – С)x2 + (D' — D)×3, and if x is made less than any assignable quantity, A — A' will be less than any assignable quantity. That is, Hence, Bx+ CÃ2 + Da3 + ··· = B'x + C'x2 + D'Ã3 + · Dividing by x and transposing, we have B — B' = (C' — C') x + (D' − D) x2, and by the same proof as above, and so on. B-B' = 0, or B=B', .... Hence, as a finite series is always convergent, it follows that the coefficient of the like powers of x in the two finite series A+ Вx + Сx2 + Dã3 + · A' + B'x + C'x2 + D'1⁄2·3 + ..... are equal if the two series are equal for every value of x. 1 .... 359. Performing the operation indicated in the fraction we obtain the infinite series 1 + x + x2 + 2~3 +........ For 1-x' all values, however, this series is not equal to the fraction. 360. When x is numerically less than 1, the series is equal to the fraction, and is convergent. 361. When x is numerically greater than 1, the series is not equal to the fraction, and is divergent. Put x= , and the value of the fraction becomes 1 + 3 + 1 + 27 + 8/1 + ···· 1+1+1+ Here the value of the sum of the first four terms gives 114; of the first five terms 10. Continuing this process further, we find that the successive sums will approach but never reach 11. Put x= = 3, and the value of the fraction becomes 1+3+9+27+81 +.... Here the value of the sum of the first four terms is 40; of the first five terms, 121. Continuing this process further, we find that the successive sums are larger and larger, and the fraction and series are not equal, Put x=1, and we have 1+1+1+1+.... Hence the value of the series can be made to exceed any assigned value, and the series is not equal to the fraction. Hence, when = 1, the series is divergent. Put x=- - 1, and we have 1-1+1-1+1-1+... 362. Hence, the value of the successive terms of the series is either 1 or 0, as the number of the terms is odd or even, and the series is not equal to the fraction. 363. From the above reasoning, it is evident that an infinite series cannot be used for the purposes of demonstration, unless it is convergent. That is, when a lies between 1 and 1. 364. If the value of each term of a series, which is alternately positive and negative, is less than the preceding term, the series is convergent. (1) (2) V4 is Now by hypothesis each of the expressions v1V2, V3 positive as is evident from (1); and it is also evident from (2) that the sum of any number of terms is less than v1. Take for example the series The sum of the first four terms is; of the first six terms, as its limit. EXPANSION OF FRACTIONS BY MEANS OF INDETERMINATE COEFFICIENTS. .. 1 − x = A + (B − A)x + (C − B + A)x2 +(D-C+B)2 + •.•. 365. For the purposes of demonstration it is necessary and sufficient that the assumed equation in its simplified form shall have all the powers of x in the right-hand member that are found in the left-hand member. The coefficient of any power of x in the right-hand member, not in the left-hand member, will vanish—as shown in the above example-and the method still applies. Any power of x in the left-hand member not in the righthand member leads to an absurdity. If we assume in the example given above we would have no term in the right-hand member corresponding to one in the left-hand; hence in equating, 1 = 0. EXERCISE 113. Expand to five terms in ascending powers of a: 366. The Theorem of Indeterminate Coefficients enables us to express a given fraction as the sum of two or more partial fractions whose denominators are factors of the denominator of the given fraction, and whose numerators contain no power of x, when the factors of the denominator are of the first degree of x, and the numerator is of a lower degree than the denominator. |