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OF THE DIFFERENTIAL METHOD.

176. THIS is a method of summing series, &c. by means of the successive differences of their terms.

Let 17 28 84 210 462 924 1716, &c. be series of numbers; Then taking the difference of the first and second, of the second and third, of the third and fourth, &c. and again the differences of those differences, and so on, we shall have the following orders of differences:

1 7 28 84 210 462 924 1716 &c.

1st. order of differences 6 21 56 126 252 462 702

35 70 126 210 330

2d. order

3d. order

4th. order

5th. order

6th. order

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Or suppose a, b, c, d, f, g, &c. to be a series; then

1st. order of differences b-a, c-b, d—c, f—d, g—f, &c.

2d. order............ c—2b+a, d—2c+b, ƒ—2d +c, g −2f+d, &c. 3d. order............ d-3c43b-a, f-3d+3c-b, g-3/+3d-c, &c. f-4d6c-46 + a, g −4f+6d —4c+b, &c.

4th. order

5th, order

g—5f+10d-10c+5b—a, &c.

177. Let D', D', D", D", D', &c. denote the first terms of the several orders of differences, respectively,

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Then by transposition we get the values of b, c, d, &c,

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But 2b — a=b+(b—a) = a + D′ + D′ = a+ 2D′, (because b = a +D′ and b—a = D′) therefore c=a+ 2D'+D".

Also, since 3b-a-3 (a+D′) +a;

we have 3c—3b+a=3 (a+2D′+D")—3 (a+D′)+a=a+3D′+3D", whence d=a+3D' + 3D" +D'''.

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Ilence it appears that the coefficients of a, D′, D", D &c. in the expression for the (n + 1) th. term of the series a, b, c, &c. are the coefficients of a binomial raised to the nth. power; that is, the (n + 1) tk. term

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Thus, for example, if the number of terms be 5, or n=5, the 6th. or

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or a +5D2 + 10D" &c. the value of g the 6th. term.

Therefore substituting n + 1 for n, the nth. term of the series a, b, c, &c.

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163

178. A general expression for the sum of any number (n) of the terms of the series, is readily obtained from the aggregate sum of the perpendicular columns as they stand in the expressions (A):

Thus, the coefficients in the columns a, a, a, &c. D', 2D1, 3D", &c. are the several orders of figurate numbers (141):

Now the sum of a+a+a+&c. to n terms is na: of D'+2D′+3D2 + &c. to n-1 terms is n.

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&c.

'—'D': (144)

2

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And the aggregate must be the sum of n terms of the series a+b+c+&c.

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-2 n-3

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D' + "

D +" .

2

2

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When the differences are finallyo, any term, or the sum number of the terms may be accurately determined; but if the differences do not vanish, the result is only an approximation: this approximate value however, will become nearer and nearer the truth as the differences diminish.

Examples.

1. What is the 17th. term of the series 1, 3, 6, 10, 15, &c. ?

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=2, p=1; these being substituted in the expression

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" + (n − 1) D' + ' —-—-— ' • n-2 D11 &c. give 1 + (n − 1) × 2 +

n

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2. To find the nth. term of the series of rectangles 1×2,

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Here a=2, D'= 10, D"=8

(”—1) 10+”—'

And 2 + (π − 1) × 10 +"="x 8=4n2-2n the required term.

3. To find the sum of n terms of the series of cubes 13 + 23 +33 +43 + &c.

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Hence it appears that the aggregate of any number of the series of cubes 13+23+33 +43 &c. taken in succession from 1, is a square number.

4. When the series is descending, the differences will be alternately minus and plus. Thus, to find the sum of the liquadrates 10+ 94 + 8+ + &c. to 8 terms:

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Here a=10000, D'=—

+24, n=8,

II

III

IV

3439, D' +974, D=204, DTM —

And 10000 x S-28 × 3439 +56 × 974–70 × 204 +56 × 24=25316 the sum required.

And in the same manner, the sums of series of higher powers may be determined.

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If we suppose n to be infinite, all its inferior powers may be rejected as inconsiderable in respect of the greatest or highest power, because any power any power of an infinite quantity is the next inferior power taken an infinite number of times, and we shall get an expression for the sum of an infinite series of powers whose roots are in arithmetical progression, having an infinite or indefinitely small quantity for the common difference:

Thus, rejecting 3n+n in the expression for the sum of a series of 2n 3 n squares, gives the sum of an infinite series of squares pro

6

or

3

3

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