RESOLUTION OF FRACTIONS INTO SERIES.

134. An Infinite Series consists of an unlimited num. ber of terms which observe the same law.

The Law of a Series is a relation existing between its terms, such as that when some of them are known the others may be found.

Any proper algebraic fraction, whose denominator is a polynomial, may, by division, be resolved into an infinite series.

Resolve the following fractions into infinite series:

;=1—p2+pt—p® +, etc., to infinity.

α

When the two terms of a fraction are finite determin

ate quantities, the fraction has necessarily a finite determinate value, which is the quotient of a divided by b.

Let us now examine the cases where the numerator or denominator, or both, reduce to zero.

While the denominator b is a constant number, if the numerator a diminishes, the value of the fraction diminishes.

Thus, in the fractions, §, %, and 1, each is less than the preceding. Hence, as the numerator a diminishes and approaches to zero, the value of diminishes and approaches to zero; and finally, 0

α

Ъ

when a=0, the expression reduces to zero.

If the numerator a, of a fraction, remains constant, and the denominator diminishes, the value of the fraction in

creases.

8

8 8

Thus, in the fractions, &, 8 1,1,1, etc., each is greater than the preceding, the values being 2, 4, 8, 16, 32, etc. Hence, when the denominator finally decreases to zero, the value of the fraction is greater than any assignable quantity; that is, infinitely great, or infinity. This is designated by the sign oo; that is,

α

0 0

137. To prove that is indeterminate in value.

If we divide both terms of any fraction, as, continually by any number, as 2, we shall have 1, 2, 1, etc.; and however long this may be continued, the value will still be. When the division is performed an infinite number of times, the fraction will assume the form 8. Hence, in this case, 8=!.

If we had taken the fraction 20, and divided as before, we should have had, finally, 8=5; and so for any other fraction.

As the value of any quantity may be expressed by a fraction, and that fraction reduced, in a similar manner, by the division of its terms, to the form 8; therefore, 8= any quantity whatever.

0 0

It is important, however, to observe, that the form is often the result of a particular supposition, when both terms of a fraction contain a common factor.

a2_b2 Thus, if x=a-b'

and we make b=a, it becomes

a2-a2 0

a-a 0'

but if we cancel the common factor, a-b, and then make bɑ, we have x=2α.

0

be

These examples show, that if the value of any quantity is fore we decide that it is indeterminate, we must see that the apparent indetermination has not arisen from the existence of a factor, which, by a particular supposition, became equal to zero.

α ∞

By similar methods, it may be shown that =0,

α

Co=1, and that 0X∞ or are indeterminate in value.

∞ ∞

0m=0,

138. Theorem.-If the same quantity be added to both terms of a proper fraction, the new fraction resulting will be greater than the first; but if the same quantity be added to both terms of an improper fraction, the new fraction resulting will be less than the first.

Let m represent the quantity to be added to each term

α

of any fraction, as then, the resulting fraction is

a+m

b+m

Since the denominators are the same, that fraction which has the

greater numerator is the greater. Now, if

α

is a proper fraction,

or if a is less than b, the second fraction is obviously greater; but if it is improper, and a greater than b, the second is less than the first; which proves the theorem.

139. Theorem.-If the same quantity be subtracted from both terms of a proper fraction, the new fraction resulting will be less than the first; but if the same quantity be subtracted from both terms of an improper fraction, the new fraction resulting will be greater than the first.

Let m represent the quantity to be subtracted from each

a

term of any fraction, as; then, the resulting fraction is

Reducing these fractions to a common denominator, we have

Reasoning as in the preceding theorem, when the original fraction is proper, the second fraction is evidently less than the first; when improper, it is greater.

5. Prove that the sum or difference of any two quantities, divided by their product, is equal to the sum or difference of their reciprocals.

6. If

+

+

c2+h2

-1,

(a—b) (a—c) (b—a) (b−c) (c-a) (c-b) prove that when the terms are multiplied respectively by b+c, a+c, and a+b, the sum =0; and that when multiplied respectively by bc, ac, and ab, it is =h2.

IV. SIMPLE EQUATIONS.

DEFINITIONS AND ELEMENTARY PRINCIPLES.

140. An Equation is an algebraic expression, stating the equality between two quantities. Thus,

x-5-3

is an equation, stating that if 5 be subtracted from x, the remainder will be 3.

141. Every equation is composed of two parts, separated from each other by the sign of equality.

The First Member of an equation is the quantity on the left of the sign of equality.

The Second Member is the quanity on the right of the sign of equality.

Each member of an equation is composed of one or more

terms.

142. There are generally two classes of quantities in an equation, the known and the unknown.

The Known Quantities are represented either by numbers or the first letters of the alphabet; as, a, b, C, etc.

The Unknown Quantities are represented by the last letters of the alphabet; as, x, y, z, etc.

143. Equations are divided into degrees, called first, second, third, and so on.

The Degree of an equation depends on the highest power of the unknown quantity which it contains.

A Simple Equation, or an equation of the first degree, is one that contains no power of the unknown quantity higher than the first.