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To find its limit, multiply numerator and denominator by the surd conjugate to V3x— a-væ+a; the fraction then becomes (3x—a)-(x+a)
1 whence by putting x=a we find that the limit is
1- 3,00 Example 3. The fraction becomes when x=1.
1-5 To find its limit, put x=1+h, and expand by the Binomial Theorem. Thus the fraction
5 Now h=0 when x=1; hence the required limit is
437. We shall now discuss some peculiarities which may arise in the solution of a quadratic equation. Let the equation be
ax2+bx+c=0. If c=0, then
ax2 + bx=0;
x=0, or 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.
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 7 to discuss the value of the other root we proceed as follows.
Write - for x in the original equation and clear of fractions;
cyo+by+a=0. Now put a=0, and we have
cy2 + by=0;
b the solution of which is y=0, or ; that is, xc=, or
Hence, in any quadratic equation one root will become infinite if the coefficient of x2 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 axa +bx+c=0, 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.
EXAMPLES XLII. Find the limits of the following expressions, (1) when x=0,
(2) when x=0. 1. (2.2-3)(3-5%)
2. 7x2 - 6x +4
34+9 3. (3+2x3)(3-5)
(x-3)(2–5x) (3x+1) (423-9)(1+)
(2x - 1)3 1--20:2 1 --X 5.
6. 2x3 - 1
(7~— 1)(x+1)3 Find the limits of 7. 33+1
when x= =-1. X2-1' 8. VX-V2a+ Væ - 2a
when x=2a. V.:2-4a2 9. (a2 — 22)+(a—)?
when x=a. (a3 —23)+(a-x)
Va + ax +32 - Val-ax+32 10.
when x=0. Va+ - va-.
CONVERGENCY AND DIVERGENCY OF SERIES.
438. WE have, in Chap. XXXVI., defined a series as an expression in which the successive terms are formed by some regular law; if the series terminates 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
U1+U2+Uz+ . + Unt 439. 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.
440. 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
1-20h 1+x+x2+28+ is
If x is numerically less than 1, the sum approaches to the
1 finite limit and the series is therefore convergent.
1 -- x' If x is numerically greater than 1, the sum of the first n
x-1 terms is and by taking n sufficiently great, this can be
-1' made greater than any finite quantity; thus the series is divergent.
If x=1, the sum of the first n terms is n, and therefore the series is divergent. If x= -1, the series becomes
1-1+1-1+1-1+. The sum of an even number of terms is 0, while the sum of an odd number of terms is 1; and thus the sum oscillates between the values 0 and 1. This series belongs to a class which may be called oscillating or periodic convergent series.
441. There are many cases in which we have no method of finding the sum of the first n terms of a series. We proceed therefore to investigate rules by which we can test the convergency or divergency of a given series without effecting its summation.
442. An infinite series in which the terms are alternately positive and negative is convergent if each term is numerically less than the preceding term. Let the series be denoted by
Un — Uz+ Uz— U4+Uz - U6 +...... where
Uy>u2>u3>44>Uz ... The given series may be written in each of the following forms: (un-u,)+(Uz-un) +(Ug-ug) +
(1), U-(Un- ug)-(44-uz)-(U8-U,) - ......(2). From (1) we see that the sum of any number of terms is a positive quantity; and from (2) that the sum of any number of terms is less than uy; hence the series is convergent. For example, in the series
1 1 1 1 1
the terms are alternately positive and negative, and each term is numerically less than the preceding one; hence the series is convergent.
443. An infinite series in which all the terms are of the same sign is divergent if each term is greater than some finite quantity however small.
For if each term is greater than some finite quantity a, the sum of the first n terms is greater than na; and this, by taking n sufficiently great, can be made to exceed any finite quantity.
444. Before proceeding to investigate further tests of convergency and divergency, we shall lay down two important principles, which may almost be regarded as axioms.
I. If a series is convergent it will remain convergent, and if divergent it will remain divergent, when we add or remove any finite number of its terms; for the sum of these terms is a finite quantity.
II. If a series in which all the terms are positive is convergent, then the series is convergent when some or all of the terms are negative; for the sum is clearly greatest when all the terms have the same sign.
We shall suppose that all the terms are positive, unless the contrary is stated.
445. An infinite series is convergent if from and after some fixed term the ratio of each term to the preceding terın is numerically less than some quantity which is itself numerically less than unity. Let the series beginning from the fixed term be denoted by
U + U2+U2t ut ..;
where r<1. Then
Un + U2+U3+04+
<u](1+r+r2 +33 + .....); that is, <ium, since r<1. .
Hence the given series is convergent.