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constant, the series of second differences is constant and the original series is of second order. In like manner it is always true that the degree of the general term is the order of the series it defines. 1 Introduction, Book I, Chapter IV. 32 2. On the Use of Differences in the Theory of Series 51. It follows that a series of powers of the natural numbers will have constant differences, as is clear from the following scheme: First Powers: 1, 2, 3, 4, 5, 6, 7, 8, . . . First Differences: 1, 1, 1, 1, 1, 1, 1, . . . Second Powers: 1, 4, 9, 36, 49, 64, . . . First Differences: 3, 5, 7, 9, 11, 13, 15, . . . Second Differences: 2, 2, 2, 2, 2, 2, . . . Third Powers: 1, 8, 27, 64, 125, 216, 343, . . . First Differences: 7, 19, 37, 61, 91, 127, . . . Second Differences: 12, 18, 24, 30, 36, . . . Third Differences: 6, 6, 6, 6, . . . Fourth Powers: 1, 16, 81, 256, 625, 1296, 2401, . . . First Differences: 15, 65, 175, 369, 671, 1105, . . . Second Differences: 50, 110, 194, 302, 434, . . . Third Differences: 60, 84, 108, 132, . . . Fourth Differences: 24, 24, 24, . . . . The rules given in the previous chapter for finding differences of any order can now be used to find the general terms for differences of any order for a given series. 52. If the general term for any series is known, then not only can it be used to find all of the terms, but it can also be used to reverse the order and find terms with negative indices, by substituting negative values for x. For example, if the general term is x2 +3x /2 and we use both negative and positive indices, we can continue the series in both ways as follows: Indices: . . . , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, . . . Series: . . . , 5, 2, 0, -1, -1, 0, 2, 5, 9, 14, 20, 27, . . . First Differences: . . . , -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, . . . Second Differences: . . . , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . 2. On the Use of Differences in the Theory of Series 33 Since the general term is formed from the differences, any series can be continued backwards so that if the differences finally are constant, the gen- eral term can be expressed in finite form. If the differences are not finally constant, then the general term requires an infinite expression. From the general term we can also define terms whose indices are fractions, and this gives an interpolation of the series. 53. After these remarks about the general terms of series we now turn to the investigation of the sum, or general partial sum, of a series of any order. Given any series the general partial sum is a function of x that is equal to the sum of x terms of the series. Hence the general partial sum will be such that if x = 1, then it will be equal to the first term of the series. If x =2, then it gives the sum of the first two terms of the series; let x = 3, and we have the sum of the first three terms, and so forth. Therefore, if from a given series we form a new series whose first term is equal to the first term of the given series, second term is the sum of the first two terms of the given series, the third of the first three terms, and so forth, then this new series is its partial sum series. The general term of this new series is the general partial sum. Hence, finding the general partial sum brings us back to finding the general term of a series. 54. Let the given series be a, aI, aII, aIII, aIV, aV, . . . and let the series of partial sums be A, AI, AII, AIII, AIV, AV, . . . . From the definition we have A = a, AI = a + aI, AII = a + aI + aII, AIII = a + aI + aII + aIII, AIV = a + aI + aII + aIII + aIV, AV = a + aI + aII + aIII + aIV + aV, and so forth. Now, the series of differences of the series of partial sums is AI - A = aI, AII - AI = aII, AIII - AII = aIII, . . . , so that if we remove the first term of the given series, we have the series of first differences of the series of partial sums. If we supply a zero as the 34 2. On the Use of Differences in the Theory of Series first term of the series of partial sums, to give 0, A, AI, AII, AIII, AIV, AV, . . . , then the series of first differences is the given series a, aI, aII, aIII, aIV, aV, . . . . 55. For this reason, the first differences of the given series are the second differences of the series of partial sums; the second differences of the former are the third differences of the latter; the third of the former are the fourth of the latter, and so forth. Hence, if the given series finally has constant differences, then the series of partial sums also eventually has constant differences and so is of the same kind except one order higher. It follows that this kind of series always has a partial sum that can be given as a finite expression. Indeed, the general term of the series 0, A, AI, AII, AIII, AIV, . . . , or that expression which corresponds to x, gives the sum of the x - 1 terms of the series a, aI, aII, aIII, aIV, . . . . If instead of x we write x+1, we obtain the sum of x terms which is the general term. 56. Let a given series be a, aI, aII, aIII, aIV, aV, aVI, . . . . The series of first differences is b, bI, bII, bIII, bIV, bV, bVI, . . . , the series of second differences is c, cI, cII, cIII, cIV, cV, cVI, . . . , the series of third differences is d, dI, dII, dIII, dIV, dV, dVI, . . . , and so forth, until we come to constant differences. We then form the series of partial sums, with 0 as its first term, and the succeeding differences in the following way: Indices: 1, 2, 3, 4, 5, 6, 7, . . . Partial Sums: 0, A, AI, AII, AIII, AIV, AV, . . . Given Series: a, aI, aII, aIII, aIV, aV, aVI, . . . 2. On the Use of Differences in the Theory of Series 35 First Differences: b, bI, bII, bIII, bIV, bV, bVI, . . . Second Differences: c, cI, cII, cIII, cIV, cV, cVI, . . . Third Differences: d, dI, dII, dIII, dIV, dV, dVI, . . . The general term of the series of partial sums, that is, the term correspond- ing to the index x, is (x - 1) (x - 2) (x - 1) (x - 2) (x - 3) 0+(x - 1) a + b + c + · · · , 1 · 2 1 · 2 · 3 but this is also the series of partial sums of the first x-1 terms of the given series a, aI, aII, aIV, . . . . 57. Hence, if instead of x - 1 we write x, we obtain the series of partial sums x (x - 1) x (x - 1) (x - 2) x (x - 1) (x - 2) (x - 3) xa + b + c + d + · · · . 1 · 2 1 · 2 · 3 1 · 2 · 3 · 4 If we let the letters b, c, d, e, . . . keep the assigned values, then we have Series: a, aI, aII, aIII, aIV, aV, . . . General Term: (x - 1) (x - 2) (x - 1) (x - 2) (x - 3) a +(x - 1) b + c + d 1 · 2 1 · 2 · 3 (x - 1) (x - 2) (x - 3) (x - 4) + e + · · · , 1 · 2 · 3 · 4 Partial Sum: x (x - 1) x (x - 1) (x - 2) x (x - 1) (x - 2) (x - 3) xa+ b+ c+ d+· · · . 1 · 2 1 · 2 · 3 1 · 2 · 3 · 4 Therefore, once a series of any order is found, the general term can easily be found from the partial sum in the way we have shown, namely, by combining differences. 58. This method of finding the partial sum of a series through differences is most useful for those series whose differences eventually become constant. In other cases we do not obtain a finite expression. If we pay close attention [ Pobierz caÅ‚ość w formacie PDF ] |