1 7, { + (7) ). a, cos O. x+. .
1 - (7)1 = , Kin 0 x + the series resembling those in (2). Further varieties will appear in Tavi.on'a Tut EOR 5. If ego) +0). x+40(2). xs+ .... be the development of a per fectly oontiuuous function of x, and if on be a function which never becomes infinite for any real value of n, positive or negative ; then the mine function may be also developed into -1(-1). -2). - A" - . . (D. C., p. 560).
O. If gam a + . and if p . . . p be the n nth boors of unity, than (D. C., p. 319) itr(p,x)+ +14.4 = +. and so on. Also if we make p p,, fic., the nth roots of and use the name results, only altering the multipliers into &c., &c., we have the sums of the same seriea with the terms alternately positive and negative.
7. One of the simplest modes of actually finding a finite expression for a series, finite or infinite, the of which are values of a rational and integral function, is the continual multiplication by 1-x, which must at last produce a finite expression. It must be remem bered that multiplication by may be performed by letting the first coefficient remain, and diminishing every other coefficient by its predeceasor. Thus a + bx+ + + .... multiplied by 1-x gives a + (6 - a) x + - + (e- c) SS+. And a finite series must be posed to be continued ad infinitum with vanishing coefficients. For example, it is required to find a finite expression for + .; write this as in the first line, and make successive multiplications by 1-x, in the following lines : 1 1 1 1+ 5x+ 6.0+ &vs+ 1 + 4x+ xs-1- Ors+ After four multiplications, then, by 1-x, the series becomes 1 +4x + whence its value is Independently of the modes of deriving series obtained from Taylor Theorem, and of which we arc to speak elsewhere, there are two modes of forming them which deserve attention. The first depends upon the numbers called the differences of nothing [Noeurso, DIFFERENCES OF]; the second on those called Numnens OF BERNOULLI.
By the first-of these any function of e can be expanded in powers of .r.
f i = f 1 + f (1 + 0 x + f (1 + . +...
Here/ (1 + 1) 03 is a symbol of the calculus of operations [OPE RATION , which expanded is Ao f 1 . + f '1 . A 03 + . . + 1 2 . 3 .
it being unnecessary to go further, because A'"0" = 0 whenever in is greater than n. (1). C., 307.) The numbers of Bernoulli occur fint in the development of (e - a series the importance of which can only be estimated by its use iu SUMMATION. Taking the numbers from the article cited above, or making 1 1 1 1 = ' " 133= B1 57' &C.
1 1 1 ip,x e - 1 2-- 7; 4. 2 - TiT + [6] - 1 3a,x 15n,xs G3n,xs + 1 = 2 + [4] [6] + " where [a] means 1 . 2 . 3 .... -1 . n.
We shall now give a number of series which are not of very frequent use, but which may sometimes be sought in a work of refer ence. Under the last predicament wo can never suppose that any of the Wowing developmente would come : (1 +x), e , a , log (1 + r), 1 +x log , sin x, or cos .c. Some terms are given; and t .... ie
omitted to save room.
xs 2.1s 62.r' 1382x" tan + 15 + 315 + 2885 + 155925 1 x xs les x7 cot x x - 3 - 5.0 61xs 277x' see =I + + 24 + 720 + 8064 1 x 7.es 31x* ewe x= - + + + + - - x 6 360 15120 6018001 xs 1 . 3 Xs 1 . 3 . 5 sin-le=x+ 2 3 + 2.4 5 + 2.4.6 7 xs - tan = x- x xs sin x = 6 + 180 + 2835 + 37800 + 407775 rel xs 17x" cos x = + + 48 + 2520 + 14175 log _tan x x PO + 2835 + 2700 + 467775These logarithms are the Naperian logarithms, as is always the case in analysis, unless the contrary be expressed (as it is usual to say, but it really never happens): 1 1 1 x x2 19xs log (1 + x) x + 2 12 + 24 720 + 160 - We must again remind the reader that the symbol +, &c., or -, &c., is throughout omitted to save room.
There is one property of series which deserves particular notice (D. C., pp. 226 and 649) as creating a most remarkable distinction between those which have all their terms positive, and those which have them alternately positive and negative. The former, oven if the diminish without limit, are not necessarily convergent ; thus 1 + .... is divergent. But if the terms be alternately positive and negative, and diminishing without limit, the series is always convergent, and the error made by stopping at any term is less than the first of the terms thrown away. And the most remarkable part of the property is that this last is true, even if the series become divergent, by having its terms increasing without limit, instead of diminishing; so that if the terms diminish for a time, and then begin to increase, the portion of the series during which the diminution takes place may be made use of in approximating to its arithmetical value. That is, if a, be all positive quantities, and if the infinite series A, + carried as far 04 a. , the error is less than whatever the law of the terms may be, or however rapidly they may afterwards increase. Let us take, for instance, 2 2.3 2.3.4 2.3.4.5 1 - + - + x xs ' Let x be ever so great, the rapid Increase of the numerators must still make this series ultimately divergent. Nevertheless, if x be con siderable, the first terms diminish so rapidly that, with the aid of the above theorem, a good approximation may be made to the arithmetical value of the function from which the series was derived. Let x=100, whence the series becomes 1 - '02 + '0006 -'000024 + '00000120 After the hundredth term the terms will begin to increase, and more and more rapidly ; but the theorem enables us, when x=100, to make the following assertions ; first, 1 Is too great, but not by so much as '02; 1-'02, or T8, is too small, but not by so much as '0006; +'0008, or '9806, is too great, but not by so much as .000024; '9806 -.000024, or '980576, is too small, but not by so much as 00000120; '980576+ '00000120, or '0805772, is too great, but not by so much an the next term ; and so on.