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tions, we shall obtain the values of the assumed coefficients as follows:

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If we substitute these values of A, B, C, etc., in equation (1), we shall have the value of x in terms of a, b, c...., and the powers of Y; that is, the given series will be reversed.

2. Given y = ax+bx+cx+dx'+ex+...., to find the value of x in terms of y.

Assume x = 4y+By® + Cy+Dy'+Ey +.... (1)
Proceeding as before, we shall obtain,

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In the preceding examples the letters a, b, c,...., represent any coefficients whatever. Hence, in reverting any series in either of these forms, we may determine the values of the assumed coefficients by an application of formula (F), or (G).

3. Revert the series y = x+2.c* +40 +8.c*+.... Assume, x = Ay+By+Cy' + Dy +....

If we now substitute in formula (F"),

a=1, b = 2, c= 4, d = 8, we shall obtain A= 1, B= -2, C=4, D= -8. Hence,

x=y-2y'+4y'-8y +...., Ans.

EXAMPLES FOR PRACTICE.

Ans. y =

1.2.3.4.5+....

Ans. 3 =

1. Revert the series y=x+x+x+*+'+....

Ans. x = y-yo+y-y+y-.... 2. Revert the series y = x+3x+5x+7x*+9x' +....

Ans. x=y^3y +-13y_674 +381y 3. Revert the series x=y

y yo
+

+
2 3 4 5

X*
at
1.2

+
1.2.3

+
1.2.3.4

+
4. Revert the series y =x-+x-x'+-" +....

Ans. x=y+y+2yo+5y'+14yo to... 5. Revert the series y = 2x+3x+4x+5x'+....

y

3y 19y 152y?
2
+

+....

16 128 1024
6. Revert the series x = 2y+4y' +6y'+8y* +10y + ...

2cm
5x"

7x* 21x
Ans.
y=
2

+
2

8
+

16 384. One of the principal objects in reverting a series is, to obtain the approximate value of the unknown quantity when the sum of the series is known. Thus, 1

6x 8x*
1. Given
4

+
3 5

t.... to find the ap

7 proximate value of x.

4x 6x9

8x* Let us put

+

t.

(1) 3

5 and consider x and s as variables. Reverting (1), by formula (F"),

so 1359 5s*
tät +
2
360 1512

(2) Now if we put s = in this equation, the result will be a con

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4x

= 2x

8 = 2x

7

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S

verging series; and we may find the approximate value of x, by computing the values of the terms separately. Thus,

1 1

.125000
2 4.2
1 1

= .010416
6 16.6

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x+

1. Given =53—20x* +80x'— 320x4 +1280x'-...., to find the approximate value of x.

Ans. x=.117647. 1

ca ace 2c* 26 2. Given + + +

to find 2

3.4 2.3

4.5 5.6 the approximate value of x.

Ans. .454620. 1 uc

2 3. Given

to find the approxi 5

6 40 112 mate value of x.

Ans. x=.201369. 1

3x 52" 7. 4. Given +

to find the ap4

2.4

4.6 6.8 proximate value of x.

Ans. x = .274655.

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+

SUMMATION OF INFINITE SERIES.

385. The Summation of a Series is the process of obtaining a finite expression equivalent to the series.

386. The method of summing a given series must always depend upon the nature of the series, or the law governing its development. Formulas have already been given for the summation of arithmetical and geometrical series. We will now investigate the methods of summing a variety of other series.

RECURRING SERIES.

387. A Recurring Series is one in which a certain number of consecutive terms, taken in any part of the series, sustain a fixed relation to the term which immediately succeeds. Thus,

1+4x+11x*+34.x*+101.c* to... is a recurring series, in which if any two consecutive terms be taken, the product of the first by 3x2 plus the product of the second by 2.x, will be equal to the next succeeding term. The coefficients of these multipliers, or

3, 2, are called the scale of relation. In the recurring series

1+x+3x++8.4° +17** +42x+100xo to... the scale of relation is 3, 2, 1.

388. A recurring series is said to be of the first order, when each term after the first depends upon the term which immediately precedes it. The scale of relation will consist of a single part, and the series will be a geometrical progression.

A recurring series is said to be of the second order, when each term after the second depends upon the two preceding terms; the scale of relation consists of two parts.

A recurring series is said to be of the third order, when each term after the third depends upon the three preceding terms ;

'the scale of relation consists of three parts.

389. To find the scale of relation and the sum of a recurring series of the second order. 1st. Let a, b, c, d, represent the coefficients of

any

four consecutive terms; and let m, n, denote the scale of relation. Then from the nature of the series, we have

ma+nb = c
c?

(P)
mb+nc =d
These two equations will determine the values of m and n.

2d. To find the sum of the series, denote the terms of the series by A, B, C, etc., and let

S=A+B+C+D+E+....

(2)

S=

The series is supposed to contain the ascending powers of x, the first

power occurring in either A or B. Then because the series is of the second order, we have

C = m Ax' +n B.x
D D = mBx' +n Cx
E = m Cx? +nDx

etc. etc. etc. Adding these equations, and observing the value of Sin (1), we have

S-A-B=mx's+nx(S-A).
Whence we obtain
A+B-Anx

(0) 1-nx-mx? 390. To find the scale of relation and the sum of a recurring series of the third order.

1st. Let a, b, c, d, e, f, represent the coefficients of any six consecutive terms; also represent the scale of relation by m, n, r. Then we have,

ma+nb+rc
mb+nc+rd = e

(T)
mc+nd+re = f
These three equations will determine the values of m, n and r.

2d. To find the sum of the series, represent the terms by, A, B, C, etc., and put

S = A+B+C+D+E+F+.. (1)
Then because the series is of the third order we shall have

D = m Ax' +n Br? +r Cw
E = mB.c+nCx+rD.

(2)
F = m Cr+nDictr Er

etc.,

etc. By addition, observing the value of Sin (1), we have

S-A-B-C = mx S+nx'(SA)+rx(S—A-B) whence we obtain

A+B+C—(A+B)rx— Anx*
S

(n) 1-rx-nxo_mx In like manner formulas may be obtained for the summation of recurring series of higher orders.

= d

etc.,

etc.,

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