457 



SERIES. 



SERIES. 



168 



ad injiintum, the last words being used strictly in the above sense, 

 without reference to any particular value of a, and only as an object 

 of algebraical operation. To what finite function of a is this an 

 algebraical equivalent in all matters of operation ? Let us consider first 

 merely results of operation, without any question as to whether the 

 series operated on have values or not, or whether expressions which 

 appear to be the same so far as operations are concerned, are to be the 

 game in value or not, when any difficulty arises as to the value of either. 

 We assume those five rules of operation and their consequences, on 

 which [OPERATION] the technical part of algebra is founded. If we 

 then call the preceding series p, we find that P and 1 +a P are the same 

 series. If then P = l + oP, we find P=l : (1 o), a result which is 

 certainly not true in any arithmetical point of view, when a>l ; for 

 in such a case the series is infinite, and the finite expression negative. 

 Leaving this, let us assume, for trial if the reader pleases, the equation 

 1 : (1 o) = s u o' ; in this change a into 1 : a, and add, which gives 



iz; + ;ZT = 2 + V + * + .- 



or 1 +s, (a* + a-*) = 0, a result which is again perfectly incongruous 

 in an arithmetical point of view. At full length it is 



To tat this curious result, by operations merely, call it <f>a, and 

 multiply A + B<H- ca-+ .... by it : the result, by common rules, will 

 be found to be 



(A + B + c + ..... ) <t>a = (A + Ba + ....) fa ; 



a result which agrees perfectly with 0a = 0, and with no other sup 

 position whatsoever. 



A great many other instances might be given, in which the use ol 

 <pa makes sense, so to speak, of results in the formation of which 

 <t.i has been used. And it is generally admitted that divergent series 

 are found to make sense, in the same manner, of almost every result in 

 the formation of which they are used ; and also that when such results 

 happen themselves to be free of divergency, there is very rarely any 

 distinction, as to either truth or clearness, between them and the 

 results of ordinary algebra ; insomuch that the objection of those who 

 would avoid them altogether, as usually stated, amounts, so far as 

 operations are concerned, to the assertion that they sometimes give false 

 results. 



If we then compare the position in which we stand with respect to 

 divergent series, with that in which we stood a few years ago with 

 respect to impossible quantities, we shall find a perfect similarity 

 The divergent series, that is, the equality between it and a finite 

 expression, is perfectly incomprehensible in an arithmetical point ol 

 view; and BO was the impossible quantity. The use of divergent 

 series has been admitted, by one on one explanation, and by another 

 on another, almost ever since the commencement of modern algebra , 

 and so it was with the impossible quantity. It became notorious that 

 such use generally led to true results, with now and then an apparent 

 exception, which most frequently ceased to be such on further 

 consideration ; this is well known to have happened with impossible 

 quantities. In both cases these apparent exceptions led some to deny 

 the validity of the method which gave rise to them, while all were 

 obliged to place them both among those parts of mathematics (once 

 more extensive than now) in which the power of producing results hac 

 outrun that of interpreting them. But at last came the complete 

 explanation of the impossible quantity [ALGEBRA], showing that all the 

 difficulty had arisen from too great limitation of definitions; am 

 almost about the same time arose that disposition among the con 

 tinental writers, of which we have spoken above, namely, to wait no 

 longer for the explanation of the true meaning of a divergent series 

 but to abandon it altogether. But why should the divergent series 

 of all the results of algebra which demand interpretation, be the onl; 

 one to be thrown away without further inquiry, when in every othe; 

 case patience and research have brought light out of darkness. 



Ho far as the matter has yet gone, very little has been done toward 

 the interpretation of a divergent series independently of its invelop- 

 ment, or function from which it is developed. When this invelop 

 ment is known, and the series deduced from it, there are means o 

 stopping the divergency, by arresting the development at any given 

 point, and turning the remainder, not into a further development, bu 

 into a finite form. Thus if $x, a given function of x, should give 

 divergent series A,, + A,* + ..... ,all that part of the developmen 

 which follows A. x* may be included in 



: 1.2.3 



is in all cases the development of (1 x)~ l , whether it be convergent 

 r divergent. Even those who reject divergent series altogether, 

 hough they would call this series, when z>l, a false or inadmissible 



development of (1 x)-', would not, though they reject it, look upon 

 t as possible to arise from any other function. But the series 



This will be proved in TAYLOR'S THEOREM, and it is a result of gre% 

 importance, because it gives the means of removing all the doubtfu 

 points of divergent series from the ordinary branches of mathematics. 



Next to the question of convergency or divergency, comes that n 

 continuity or discontinuity. We are not here speaking of continuit 

 of value, but of form. A series is continuous when for all values of . 

 it represents the same function of r. Thus sjc* or yfl + x { + y? + . . . 



L sin n6 

 n 



or sin 9 



siu 29 



sin 39 



s discontinuous ; for certain values of 9 it represents one function, 

 and for other values another. When 9 is any multiple of TT [ANGLE] it 

 = 0; when 9 falls between - T and + ir, it is 49 ; when 9 falls 

 Between * and Sir, it is 49 *, &c. ; in fact, it stands for 49 mir, 

 where m is to be so taken that 49 nnt shall fall between 4* and 

 i- Jir. Again, the series 



a'x 1 x 



B (l + a-xHl+a"- 1 -'*) - (a ~ 1) (* + 1) (1 -a) (x + 1) 



according as a > 1 or < 1 : and when a = 1, it is infinite. Remember 

 that by calling a series infinite we do not merely mean that it is 

 divergent, for a divergent series may be the development of a finite 

 quantity ; thus 1 + 2+4+.. ..is a development of - 1 from the 

 form (1 2) -'. But we mean that the arithmetical value of the 

 function developed is infinite when we say that the series is infinite. 



Discontinuity of form may be in many cases avoided by an extension 

 of the modes of algebraical expression. Thus if we write down the 

 expression 



JT^"l lx + 1 



and consider I as having a very great value, the second term will be 

 very small or very near to unity, according as a is > 1 or < 1. If we 

 introduce the symbol o* as representing <x> when a > 1, and when 

 o < 1, we have, on putting for i, the representation of both forms 

 of the preceding series in one. We shall now proceed to point out 

 some of the principal modes of transforming series into others, or 

 deducing others from them, so far as this is done without interfering 

 with the developments in TAYLOR'S THEOREM. 



1. If 0x can be developed into a + a,x + OjX 2 + . . ., then aft +a 1 b > x + 

 a b-x* + . . . can be developed as follows (D. C., p. 239). Let the last 

 be ifcc, and from ft, ft,, 6,, &c. [DIFFERENCE] form Aft, A 2 ft, &c. : then 



A 2 ft /( A 3 ft , , 



where 0'x, 0"x, &c., are the euccessive differential coefficients of 0./> 

 If ft, 4 , &c. be values of a rational and integral function of , denoted 

 by ft. the preceding is not an infinite series, but a finite expression. 

 We have not room for examples, and it is to be remembered that this 

 is an article of reference. Particular classes of instances are 



4 Aft.* A'ft.x 2 



4 + ft,x + .-.. , 



n-1 



(1 - *) 

 n-1 n 



(I-*) 3 



- 2 



r x - 1 



{6 + A4 i^ + 



, 

 A* 



"1 

 } 



(1 +)- 



b + 6,* +fr. 



the preceding is a case of more general theorem (D. C., p. 565) fn m 

 which the following may also be deduced : 



b V ft . x V'ft.a 8 



b + ft,* t .. . = jf^T^ + (1 + ,.,.)!- + (1 + mxf + " ' 



yi = i, + mb, V-b = b, + 2mft, + mfb, &c. By this theorem many 

 divergent series may be converted into convergent ones, or the con- 

 vergency of convergent series may be increased. 



x sin 



2. Letr= V(l-2cos.a: + x?), tan $ = _ x coa g 



Then b + ft, cos 9. x + ft, cos 29. -r" + b, cos 38. x 3 + . . . = 



b A*-* 



cos -j + cos ( + 



+ cos (29 + 30) 



+ . 



and ft, sin fl x + b, sin 29 . x> + ft 3 sin 39. x 3 + . .. = 

 }- + an (9 + 20) ^ + sin (29 + 30) ^ + 

 3. Let 0x be a rational and integral function of x ; then 

 0x + (x + 1) + (x + 2) . a? + . . . = j-^-jj 0x + 



r^* + ~(T=d)> ~T f (l-a) 2.3 

 A, = a + 4a* + a', A, = a + llo 2 + lla 3 + a 



A, = o + 26a" + 66a 3 + 28a< + a 5 , &c. 

 A. = a + 57a" + 302 (a + o 4 ) + 67a s + a". 



