Clearing of fractions, we have

2-2x+4x2

= (Ax + B) (1 + x2) (1 − x) + (Cx + D) (1 − x) +E (1 + x2)2 =(A+E) + (AB) 3+ (−A+B-C+2E) x2

+(A-B+C-D)x+(B+D+E).

Equating coefficients of like powers of x, we obtain

- A+E = 0, A-B=0,- A+B-C+2 E = 4,
A-B+C D = −2, B+D+E=2;

whence,

A=1, B=1, C=-2, D=0, E=1.

case, as we have seen, But if n be not a posi

when n is a positive integer. In this the series ends with the n + 1th term. tive integer, the expression on the right of (1) will continue without end, since no factor of the form n k can reduce to 0. Therefore the series will have no meaning unless it be convergent.

2. It is proved in Elements of Algebra, Ch. XXXI., that this series is convergent when a lies between 1 and +1; and, in Ch. XXXII., that when the series is convergent, it is the expansion of (1+)".

This infinite series can be taken as the expansion of (1 + x)3, = √(1 + x) only when a is numerically less than 1.

3. Expansion of (a + b)". - We have

In a similar way it can be shown that, when a is numerically

n (n-1)
1.2

an-262 +

Notice that when n is a fraction or negative, formula (3) or (4) must be used according as a is numerically greater or less than b.

4. Ex. 1. Expand

1
(a-4b2)

to four terms.

If we assume a > 4 b2, by (3), Art. 3, we have

= ( a − 4 b2) - } = a ̄3 + ( − §) a ̄† ( − 4 b2)

1

(a− 4 b2)

If a < 4b2, we should have used (4), Art. 3.

Any particular term can be written as in Ch. XXII., Art. 9.

Ex. 2. The 6th term in the expansion of (x

is

Therefore √17 = 4.1231, to four decimal places.

EXERCISES.

(1)

+

22. 5.

Find to four places of decimals the values of : 23. √27. 24. 35. 25. 700.

26. 258.

CHAPTER XXVI.

LOGARITHMS.

1. It is proved in Elements of Algebra, Ch. XXXV., that a value of x can always be found to satisfy an equation of the form

10x = n,

wherein n is any real positive number. E.g., when n = 10, x=1, when n = 100, x2, when n = 1000, x = 3, etc.

When n is not an integral power of 10, the value of x is irrational, and can be expressed only approximately. Thus, when n24, the corresponding value of a has been found to be 1.38021, to five decimal places; or

A value of x is called the logarithm of the corresponding value of n, and 10 is called the base.

In general, a value of a which satisfies the equation b* = n, is called the logarithm of n to the base b.

E.g., since 28, 3 is the logarithm of 8 to the base 2; since 102 100, 2 is the logarithm of 100 to the base 10.

=

The Logarithm of a given number n to a given base b is, therefore, the exponent of the power to which the base b must be raised to produce the number n.

2. The relation b a is also written x= log, a, read x is the logarithm of a to the base b.

238

Thus,

and 3 log. 8,

=

are equivalent ways of expressing one and the same relation.