INFINITE SERIES, ETC.] 



MATHEMATICS. ALGEBRA. 



513 



(2). 



11 



3)(z 5) 



3 



79 



(3). 



(4)- 





 ~ 



70(x 5) 

 a 1 



(x a) (x 6) a 6 x a a 6 x i 



(x a)(x 6)(x c) (c a) (b o) z a 



I 



(a 6) (c 6)' x 6 (a c) (6 c ) ' x c 



(5). A series is a number of algebraical expressions, 

 each of which is connected with those which precede it 

 in some determinate manner. 



For example : In the treatise on Elementary Algebra, 

 we have had examples of series in the arithmetical and 

 geometrical progressions.* In the former case, each 

 term is derived from the one preceding it by adding a 

 certain known number called the common difference. In 

 the latter case, each term is derived from the one pre- 

 ceding it by multiplying that term by a certain known 

 number called the common ratio. Hence, 



a -j- or + ar 2 + ar 3 + 

 and 1 + r + r 1 + r 3 -\ ----- 

 arc series. 



ji e f. A series is called a finite series when it has an 

 Mwignable last term. It is called an infinite series 

 i, if we fix on any term whatever, there are terms 

 beyond it. 



Thus, l + r + r*H ----- +r" is a finite series. But 

 1 -)- r + r 2 4 d * n /- i 8 an infinite series, because, if 

 we take any term whatever for instance, the 60th, or 

 600th, or 5000th there are always terms beyond it. 



6. To explain what is meant by a Convergent and 

 a Divergent Scries. 



Def. If the sum of the terms of a series has an 

 arithmetical limit when the number of terms is infinite, 

 that series is convergent ; if otherwise, it is divergent. 



If we divide 1 by 1 r, we shall produce 1 + r + r 1 

 -(-.... which series we can continue to produce to any 



number of terms whatever. Hence the fraction . ' 



and the series, 1 + r + i 3 + ad / are equivalent 

 to each other ; or 



injinitum. 



- = 



Now, it has been already proved, that if r < 1, by 

 taking a sufficiently large number of terms, the numerical 

 value of the series can be made to approach to the nume- 



rical value of .. _ i to within any assignable limits. 



For instance, if r = J then . _ = 2; and if we take four 



terms the series equals 1-875. If we take five terms it 

 equals 1 '9375 ; if six terms, it equals 1 '96875 ; and 

 hence in the extreme case, when we suppose the number 

 of terms to be infinitely large, the series is actually equal 

 to 2. And hence if r be less than 1, 



. _ =>l+r + r*+r* + ad infini turn, 



where by the sign = we mean that the fraction ^ _ f 



is arithmetically equal to the series. But if r be greater 

 than 1, for instance, if equal to 2, the fraction equals 

 1 ; whereas if we take four terms, the series 15 ; if 

 five terms, 31 ; if six terras, C3 ; and so on, where there 

 is no trace of approximation towards arithmetical 

 equality between the series and the fraction. In the 

 former case the series is said to be convergent, in the lat- 



Bee ante, p. 477, it leq. 

 VOL. I. 



ter divergent ; and if we include both cases in the ex- 

 pression, 



1 



I f =1 4- r 4- "r + d injinitum ; 



it must be understood that the sign = signifies alge- 

 braically equivalent, not arithmetically equal. This 

 explanation will be sufficient to enable the student to 

 understand the meaning of the terms convergent and 

 divergent, when applied to special series. The general 

 questions that are suggested by series, and their con- 

 vergency and divergency, belong to the higher parts of 

 the science and many of them are still doubtful 



7. A test for ascertaining the Convergency of a given 



Series. 

 We have already seen that 



., =>14-*'4'' J 4-' 4 4-. is convergent when r is 



< 1. Hence if we have a series 



A+B4-C4-D+ 



and can show that B < rA, O < r^A, D < r'A, <fec. 

 Then 



A4-B4-C 4- D 4- . . . < A(l +r4-r4-r4-.. ) 

 This latter is convergent if r be < 1. And if so, the 

 former must plainly be convergent too. This gives us a 

 test for ascertaining whether a given series is convergent, 

 which we shall find useful hereafter. The student must 

 remember that, though all series which submit to this 

 test are convergent, many may be convergent which do 

 not submit to it. 



For instance, to ascertain whether the series 



0* 6 s 



" T i ..>.Q 4" I.O.Q..J g T & c -> ad injinitum, is convergent 



X A O i. ' jj O"'i*O 



when < 2. 



The series may be written 

 ,<-.*. * 



9 00. 

 -^-X-=-<^r.-^-oriB< 



23 2 



2" 2 



(I/ 



0* 9 * 



Similarly, -- . , < f^\ and so on for the other 



terms ; hence the given series is less than 



" \ 

 and this is convergent if f-^ 1 < 1, or if < 2 ; and hence 



the given series is convergent if < 2. 



N.B. j-^r-j; = 1 r4-r a ^4- .... is true arith- 

 metically when r < I. Hence, the above test of con- 

 vergency holds good when the terms are alternately 

 positive and negative. 



In the following pages we shall confine our attention 

 to three series the Binomial, the Exponential, and the 

 Logarithmic. 



8. To state the Binomial Theorem.^ 



If a. 6. n. are any numbers whatever, then 



(o 4- 6)" = O" 4" n - a " ' 



n(n-l)(n-2) , 



_I_ _^^^__^^^_^_^^^^^^_ /T W 



36' 4- &o. 



The reader will observe that the first term is *,, that 

 in each term the index of a diminishes by unity, while 

 the power of b continually increases by unity, so that the 

 sum of the indices of each term is n. 



Again, the coefficient of each term has for each deno- 

 minator the continued products 1.2.3 .... up to the 

 index of b inclusive ; and for numerator, the continued 

 product n(n 1) (n 2) . . . . down to the index of a 



+ See ante, p. 402. 



3 u 



