2x2-4x+5

2. Find the limiting value of

when x

3x2+2x2 - 1

becomes infinite.

Divide each term of the numerator and denominator by x3. Then,

As x increases indefinitely, each term that contains x of the last fraction approaches 0 as a limit (Theorem 4), and the fraction ap

CHAPTER XXIV.

PROPERTIES OF SERIES.

Convergent and Divergent Series.

390. By performing the indicated division, we obtain

from

1

1

the infinite series 1+ x + x2 + x3 + ..... This series, however, is not equal to the fraction for all values of x.

391. If x is numerically less than 1, the series is equal to the fraction. In this case we can obtain an approximate value for the sum of the series by taking the sum of a number of terms; the greater the number of terms taken, the nearer will this approximate sum approach the value of the fraction. The approximate sum will never be exactly equal to the fraction, however great the number of terms taken; but by taking enough terms it can be made to differ from the fraction by as little as we please.

1

Thus, if x = the value of the fraction is 2, and the

The sum of four terms of this series is 17; the sum of five terms, 118; the sum of six terms, 13; and so on. The successive approximate sums approach, but never reach, the finite value 2.

392. An infinite series is said to be convergent when the sum of the terms, as the number of terms is indefinitely increased, approaches some fixed finite value; this finite value is called the sum of the series.

393. In the series 1 + x + x2 + x3 + ... suppose X numerically greater than 1. In this case the greater the number of terms taken, the greater will their sum be; by taking enough terms we can make their sum as large as we please. The fraction, on the other hand, has a definite value. Hence, when x is numerically greater than 1, the series is not equal to the fraction.

Thus, if x = 2, the value of the fraction is 1, and the series is

1+2+4+8+.........

The greater the number of terms taken, the larger the sum. Evidently, the fraction and the series are not equal.

394. In the same series suppose x = 1. In this case the fraction is and the series 1+1+1+1+............

1

1 1-1-0'

The more terms we take, the greater will the sum of the series be, and the sum of the series does not approach a fixed finite value.

If x, however, is not exactly 1, but is a little less than 1,

1

the value of the fraction will be very great, and the

1-x

fraction will be equal to the series.

Suppose x = - 1. In this case the fraction is

1
is
1+1

1 2'

and the series 1-1+1-1+...... If we take an even number of terms, their sum is 0; if an odd number, their sum is 1. Hence, the fraction is not equal to the series.

395. A series is said to be divergent when the sum of the terms, as the number of terms is indefinitely increased, either increases without end, or oscillates in value without approaching any fixed finite value.

No reasoning can be based on a divergent series; hence, in using an infinite series it is necessary to make such restrictions as will cause the series to be convergent. Thus, we can use the infinite series 1 + x + x2 + x + ..... when, and only when, x lies between + 1 and

1.

396. THEOREM. If two series, arranged by powers of x, are equal for all values of x that make both series convergent, the corresponding coefficients are equal each to each.

For, if A + B x + Cx2 + ... = A' + B'x + C'x2 + ....., by transposition,

A - A' = (B' – B) x + (С" — C) x2 + .....

Now, by taking a sufficiently small, the right side of this equation can be made less than any assigned value whatever, and therefore less than A - A', if A A' has any value whatever. Hence, A - A' cannot have any value.

or

Hence, Bx + Cx2 + Dx3 + ..... = B'x + C'x2 + D'x® + .....,

(B – B) x = (C' — C) x2 + (D' – D) x3 + .....

Divide by x,

B - B' = (C' — C) x + (D' – D) x2 + .....

By the same proof as for A — А',

B - B' = 0, or B = B'.

In like manner,

C = C', D = D'; and so on.

Hence, the equation

A + Bx + Cx2 + ...... = A' + B'x + C'x2 + ....., if true for all finite values of x, is an identical equation; that is, the coefficients of like powers of x are equal.

.. 2 + 3x = A + (B + A)x + (C + B + A) x2 + (D + C + B)x3 + .....

By § 396,

whence,

A = 2, B + A = 3, C + B + A = 0, D + C + B = 0 ;

B = 1, C = -3, D = 2; and so on.

The series is of course equal to the fraction for only such values of xas make the series convergent.

NOTE. In employing the method of Indeterminate Coefficients, the form of the given expression must determine what powers of the variable x must be assumed. It is necessary and sufficient that the assumed equation, when simplified, shall have in the right member all the powers of x that are found in the left member.

If any powers of x occur in the right member that are not in the left member, the coefficients of these powers in the right member will vanish, so that in this case the method still applies; but if any powers of x occur in the left member that are not in the right member, then the coefficients of these powers of a must be put equal to 0 in equating the coefficients of like powers of x; and this leads to absurd results. Thus, if it were assumed that