275. We shall now discuss some peculiarities which may arise in the solution of a quadratic equation.

that is, one of the roots is zero and the other is finite.

If b=0, the roots are equal in magnitude and opposite in sign. [Art. 118.]

с

If a = 0, the equation reduces to bx+c=0; and it appears that in this case the quadratic furnishes only one root, namely But every quadratic equation has two roots, and in order to discuss the value of the other root we proceed as follows.

thus

b

1

Write for x in the original equation and clear of fractions;

y

Hence, in any quadratic equation one root will become infinite if the coefficient of x2 becomes zero.

2

This is the form in which the result will be most frequently met with in other branches of higher Mathematics, but the student should notice that it is merely a convenient abbreviation of the following fuller statement:

In the equation ax2 + bx + c = O, if a is very small one root is very large, and as a is indefinitely diminished this root becomes indefinitely great. In this case the finite root approximates as its limit.

to

с

b

The cases in which more than one of the coefficients vanish may be discussed in a similar manner.

a simple equation is indefinitely great if the coefficient of x is indefinitely small,

a'

=

If ab' - a'b= 0, then x and y are both infinite.

a b

In this case

= m suppose; by substituting for a, b, the second

equation becomes ax + by +

ax+by+

0.

m

is not equal to c, the two equations ax + by + c = 0 and

m

=

O differ only in their absolute terms, and being inconsistent cannot be satisfied by any finite values of x and y.

0

Here, since bc' – b'c = 0 and ca' - c'a = 0 the values of x and y each assume the form and the solution is indeterminate. In fact, in the present case we have really only one equation involving two unknowns, and such an equation may be satisfied by an unlimited number of values. [Art. 138.]

The reader who is acquainted with Analytical Geometry will have no difficulty in interpreting these results in connection with the geometry of the straight line.

275. We shall now discuss some peculiarities which may arise in the solution of a quadratic equation.

that is, one of the roots is zero and the other is finite.

If b=0, the roots are equal in magnitude and opposite in sign. [Art. 118.]

If a = 0, the equation reduces to bx+c=0; and it appears that in this case the quadratic furnishes only one root, But every quadratic equation has two roots, and in order to discuss the value of the other root we proceed as follows.

namely

с

b

1

Write for x in the original equation and clear of fractions;

y

Hence, in any quadratic equation one root will become infinite if the coefficient of x becomes zero.

This is the form in which the result will be most frequently met with in other branches of higher Mathematics, but the student should notice that it is merely a convenient abbreviation of the following fuller statement:

In the equation ax2 + bx + c = O, if a is very small one root is very large, and as a is indefinitely diminished this root becomes indefinitely great. In this case the finite root approximates to - as its limit.

с

b

The cases in which more than one of the coefficients vanish may be discussed in a similar manner.

CHAPTER XXI.

CONVERGENCY AND DIVERGENCY OF SERIES.

276. An expression in which the successive terms are formed by some regular law is called a series; if the series terminate at some assigned term it is called a finite series; if the number of terms is unlimited, it is called an infinite series.

In the present chapter we shall usually denote a series by an expression of the form

277. Suppose that we have a series consisting of n terms. The sum of the series will be a function of n; if n increases indefinitely, the sum either tends to become equal to a certain finite limit, or else it becomes infinitely great.

An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quantity however great n may be.

An infinite series is said to be divergent when the sum of the first n terms can be made numerically greater than any finite quantity by taking n sufficiently great.

278. If we can find the sum of the first n terms of a given series, we may ascertain whether it is convergent or divergent by examining whether the series remains finite, or becomes infinite, when n is made indefinitely great.

For example, the sum of the first n terms of the series

1-x"