This series can also be expanded by the binomial theorem,

h

for =h(a—b)'. This observation is applicable to seve

a-b

ral other examples.

(Art. 139.) A fraction of a complex nature, or having com

pound terms, such as

1-x 1-2x-3x2

, may give rise to an infinite series, but there will be no obvious ratio between the terms. Some general relation, however, will exist between any one term and several preceding terms, which is called the scale of relation, and such a series is called a recurring series. Thus the preceding fraction, by actual division, gives 1+x+5x2+ 13x2+41x1+121æ3, &c., a recurring series, which, when carried to infinity, will be equal to the fraction from which it is derived. 1+2x 1-x

Expand

into a series

Ans. 1+3x+4x2+7x3+11xa+18x3, &c.

CHAPTER III.

SUMMATION OF SERIES.

(Art. 140.) We have partially treated of this subject in geometrical progression, in (Art. 121); the investigation is now more general and comprehensive, and the object in some respects different. There we required the actual sum of a given number of terms, or the sum of a converging infinite series. Here the series may not be in the strictest sense geometrical, and we may not require the sum of the series, but what terms or fractional quantities will produce a series of a given convergency.

The object then, is the converse of the last chapter; and for every geometrical series, our rule will be drawn from the sixth

example in that chapter; that is,

1

a

a being the first term

of any series, and r the ratio. We find the ratio by dividing any term by its preceding term.

Hence, to find what fraction may have produced any geometrical series, we have the following rule:

RULE. Divide the first term of the series by the algebraic difference between unity and the ratio.

EXAMPLES.

1. What fraction will produce the series 2, 4, 8, 16, &c?

Here a=2, and r=2; therefore,

2

Ans.

1-2

2. What fraction will produce the series 3-927—81, &c.? Here a=3, and r=- -3; then-r=3,

3. What fraction will produce the series, Toō, &c.?

10

Ans.

1.

therefore,

5. What fraction will produce the series 1+2x+2x2+2x3, [See example 3, (Art. 138.) and the observation in connection.]

&c.

9. What fraction will produce the series

1+a—a3—a1+-a®+a2—ao—a1o, &c.?

See example 5, (Art. 138.)

2ab

Ans.

a+b

Put 1+ab; then a+aab, and a+aab, and the series becomes b-ab+ab, &c.

10. What fraction will produce the series x+x2+x3, &c.?

11. What fraction will produce the series 1++,5, &c. ?

Hence, the sum of this series, carried to infinity, is 2.

In the same manner, we may resolve every question in (Art.

121.)

12. What is the sum of the series, or what fraction will produce the series 1-a+a-a+a®—a2+a—, &c.?

(Art. 141.) We have explained recurring series in (Art. 139.) and it is evident that we cannot find their equivalent fractions by the operation which belongs to the geometrical order, as no common relation exists between the single terms. The fraction 1-+-2x by actual division, gives the series 1+3x+4x2+7x3 1-x-x2 +11x+18x3, &c., without termination; or, in other words, the division would continue to infinity.

Now, having a few of these terms, it is desirable to find a method of deducing the fraction.

There is no such thing as deducing the fraction, or in fact no fraction could exist corresponding to the given series, unless order or a law of dependence exists among the terms; therefore some order must exist, but that order is not apparent.

Let the given series be represented by

A+B+C+D+E+F, &c.

Two or three of these terms must be given, and then each succeeding term may depend on two or three or more of its preceding terms.

In cases where the terms depend on two preceding terms we may have

In cases where the terms, or law of progression depend on three preceding terms we may have

The reason of the regular powers of x coming in as factors, will be perfectly obvious, by inspecting any series.

The values of m, n and r express the unknown relation, or law that governs the progression, and are called the scale of relation. We shall show how to obtain the values of these quantities in a subsequent article.

(Art. 142.) Let us suppose the series of equations (1), to be extended indefinitely, or, as we may express it, to infinity, and add them together, representing the entire sum of A+B+ C+D, &c., to infinity, by S; then the first member of the resulting equation must be (S-A-B), and the other member is equally obvious, giving

S-A-B=mx(S-A)+nx2 S

A+B-mAx

Hence, S

1-mx-nx2

(a)

In the same manner, from equations (2), we may find

(Art. 143.) The form of a series does not depend on the value of x, and any series is true for all values of x. Equations (1) then, will be true, if we make x=1.

Making this supposition, and taking the first two equations of the series (1), we have

And

C=mB+nA)

D=mC+nBS

(c)

In these equations, A, B, C, D, are known, and m and n unknown; but two unknown quantities can be determined from two equations; hence m and n can be determined.

U