And by supposing m=m'=m"=m" . . . in which case

but

P is a whole number, therefore the second member of this equation is the development of (1+z)", which gives y=(1+%)", whence

To demonstrate this formula, for the case in which m is a negative fraction or whole number, it is only necessary to suppose, m'=-M, in the equation (3) obtained from the equations (1) and (2), for when m+m'=0, the equation (3) reduces to yy'=1; whence

But since m is negative by hypothesis, m' or —m, must be positive, and we have

The fifth term can be found by multiplying the fourth by

then changing the sign of the result, and so on.

3n-1

4n

241. REMARK. When the terms of a series go on decreasing in value, the series is called a decreasing or converging series; and when they go on increasing in value, it is called a diverging series.

In a converging series the greater number of terms we take in the series, the nearer will we approximate to the true value of the proposed series. When the terms of the series are alternately positive and negative, we can, by taking a given number of terms, determine the degree of approximation.

For, let a-b+c-d+e-f+. . ., &c. be a decreasing series; b, c, d... being positive quantities, and let a denote the number represented by this series.

The numerical value of x is contained between any two consecu. tive sums of the terms of the series. For take any two consecutive sums,

a-b+c-d+e-f, and a−b+c-d+e—f+g.

...

In the first, the terms which follow -f, are g-h, + k−l+ g―h,+k―l+ but since the series is decreasing, the partial differences g-h, k—l, . . . . are positive numbers; therefore, in order to obtain the complete value of x, a certain positive number must be added to the sum a-b+c-d+e-f. Hence we have

... ·

a−b+c―d+e-f<x.

In the second series, the terms which follow +g are -h+k, -1+m Now, the partial differences -h+k, −l+m . . are negative; therefore, in order to obtain the sum of the series, a negative quantity must be added to

a-b+c-de-f+g,

or, in other words, it is necessary to diminish it. Consequently a-b+c-d+e-f+g>x.

Therefore, x is comprehended between these two sums.

The difference between these two sums is equal to g. But since x is comprised between them, their difference g must be greater than the difference between x and either of them; hence, the error com. mitted by taking a certain number of terms, a-b+c-d+e-f, for the value of x, is numerically less than the following term.

242. The binomial formula also serves to develop algebraic expressions into series.

In the binomial formula, make m=—1, x=1, and a――z, it be

or, performing the operations, and observing that each term is com posed of an even number of factors affected with the sign

The same result will be obtained by applying the rule for divi. sion (Art. 55).

or

3

2(1-2)=2(1+32+622+1023+152a+ . . . .)

To develop the expression √2z-22 which reduces to

VI(1-1), we first find

1

–1

(-1) (-)+글..()+...

1

1-2

-Z

23.

1

EXAMPLES.

1

1. To find the value of

(a+b)a, or its equal (a+b)—a in an in

finite series.

Method of Indeterminate Co-efficients. Recurring Series

243. Algebraists have invented another method of developing algebraic expressions into series, which is in general, more simple than those we have just considered, and more extensive in its applications, as it can be applied to algebraic expressions of any nature whatever.

Before considering this method, it will be necessary to explain what is meant by the term function.

Let a=b+c. In this equation, a, b and c, mutually depend on each other for their values. For,

a=b+c, b=a-c, and c-a-b.

The quantity a is said to be a function of b and c, b a function of a and c, and c a function of a and b. And generally, when one quantity depends on others for its value, it is said to be a function of those quantities on which it depends

In order to give some idea of this method of development, we will

suppose it is required to develop the expression

α

into a se.

a'+b'x

It is plain

ries arranged according to the ascending powers of x.