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(1)

The law of expansion is obvious in both quotients, and we havo from the same fraction two series; thus,

1

=l- a ta' a' ta' - ao ti
1+a
1 1 1 1 1 1 1
+ +

(2)
a> ao

a We observe that each of the series obtained is by its law of devel opment, an infinite series.

a+1

art....

a

a

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5. Convert

1

into an infinite series. 1-ata

Ans. 1ta-a-a* tao+a'-a-a" to...

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6. Convert

into an infinite series. 1-2-3x

Ans. 1+x+5x*+130° +41x* to... 1+2.

into an infinite series. 1-2

Ans. 1+3x+4x+7x+11x*+18x' +...

7. Convert

METHOD OF INDETERMINATE COEFFICIENTS.

382. It is evident that if a fraction be developed into a series, the equation which results by placing the fraction equal to its development is capable of being satisfied for any value of the unknown quantity; in other words, it is an identical equation.

On this fact depends an important method of expanding an algebraic expression into a series, called the Method of Indeterminate Coefficients. It consists in assuming the required development in the form of a series with unknown coefficients, and afterward determing the values of the coefficients by means of the known properties of identical equations.

1+2.0 1. Develop

into an infinite series. 143x

Assume
1+2.c

A+B.c+Cc+Dx"+Ex*+.... (1) 1-3x Clearing this equation of fractions, and transposing all the terms to the second member, we have 0=(1-1)+ B ++ cx+ D + E +... (2) -3A -3B

--30 -3D -2 The term A-1 may here be considered as the coefficient of co understood.

Now because equation (2) is an identical equation, the coefficients of the different powers of x are separately equal to zero, (368, IV). Thus,

A-1 = 0, whence A= 1;
B-3A-2 = 0,

5;
C3B = 0,
D-3C = 0,

D = 45;
E-30= 0,

E = 135, eto.
Substituting these values in (1) we have
1+2.c

=1+5x+15x* +45x+135x*+.... 1-3x

B =
C =

15;

or x,

1+2 2. Develop

into an infinite series. 2—2x* +6x®

1 We perceive that the first term of the series must be Therefore, assume 1+

= A.- +B.co+Cc+Dx+Ex+Fx*+.... x-2.'+6.co Clearing of fractions and transposing, we have 0 = A A xo + B | x+ C | x*+ D | x'+ E X* + Fx.. -1 —2A -2B -20 -20 -2E

+6A +6B +6C +6D Putting the coefficients equal to zero, A-1 = 0, whence, A=

1;
B--2A-1 =: 0,

B
C2B+6A=0,

c
D2C+6B=0,
E2D+60=0,
F-2E+6D=0,

F=+ 36;
G-2F+6E=0,

G=+288, etc. Substituting these values in the assumed development (1), and observing that the term containing C will disappear because C = 0,

-1

3;

0;

D=

18; 36;

we have

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1+: 1

+3–180°—360° +36x*+288x®.... -2x* +6x9 NOTE.- It is not necessary to transpose the terms to one member ; for if neither member is zero we have simply to equate the coefficients of the like powers of x in the two members, according to the third property of identical equations.

The method of Indeterminate Coefficients is applicable to a great variety of examples, but always with this provision, viz. : That we determine by inspection what power of the variable will be contained in the first term of the expansion, and make the first term of the assumed development correspond to the known fact.

If the assumed development commence with a power of the variable higher than it should, the fact will be indicated by an absurdity in one of the resulting equations. If, however, the assumed development commence with a power of the variable lower than is

necessary, no absurdity will arise ; but the redundant terms will disappear by reason of the coefficients reducing to zero.

EXAMPLES FOR PRACTICE.

1-2 1. Develop into a series.

1-3x

Ans. 1+x+3x* +9x+27x* +81x*+....

2. Develop

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1+2=

into a series. 1

Ans. 1+3x+4x' +7xo+11.* +18° +....

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3. Develop

into a series. 1-3x-2x Ans. 1+2x+8.c*+28x' +100x* +356x +....

4. Develop

x(1+52)

into a series. (1-22)

Ans. x+9x* +32.c +92 +240x*+....

5. Develop

2

into a series. 3x-2x:

2 4 8.x 16x9 32.c Ans. + + + + +....

9 3x

27 81 243

1 6. Develop

into a series. 1+20° +3:04 Ans. 1—2x*+ ** +4x°-11x +10x" +132"-....

1+x
7. Develop

into a series.
(1+ax)
Ans. 1+(1—2a)x—(20—3a%)xo+(3'—4a“)x*—..
8. Develop 11-into a series.

x® 3x8 3.5x*
Ans. 1-

2 2 : 4 2.4.6 2.4.6.8

NOTE. – Assume V1-x= A+Bu+Ca+Dic' +....; then square both members, and the equations for the coefficients will be readily obtained.

Ans. 1+7

+

+

9. Develop V1+3x+52°+72°+9x*+.... into a series of rational terms.

3x

11x 23x 179.x*
+
2

+.... 8 16 128 1–2x+3.c _Axo+5x°—6.x" +.... into an equiv10. Develop

1+ **+ **+ x + x + *"+.... alent series.

Ans. 1–3x2+52*—7xo+9x°–11x: +....

REVERSION OF SERIES.

383. The Reversion of a Series is the process of finding the value of the unknown quantity in the series, expressed in terms of another unknown quantity.

1. Given y = ax+bxo+ca'+dx+exo+ ...., to find the value of x in terms containing the ascending powers of y.

In this equation, x and y are two indeterminate quantities, and either may have any value whatever without altering the form of the series. We may therefore apply the method of Indeterminate Coefficients. Assume

x = Ay+By' + Cy+Dy' + Ey +.... (1) We may now find by involution the values of x®, x*, x*, ', etc., carrying each result only to the term containing y". Then substituting for x, x*, **, etc., in the given equations we shall have, after transposing y, 0 = aA ytaB

ac y+ a Dy+ a E yo.... bA? +26AB +26AC

+20 A D CA' + 6 BY +26BC + 3cA’B

+3cA'C
+ d 4

+ 3cA BY
+4dA'B

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+

+

e A

This is an identical equation, being true for all values of y. And if we place the coefficients of the different powers of y separately equal to zero, (368. IV), and reduce the resulting equa

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