0 This will be proved in Terton's THEOREM, and it is a result of great importance because it gives the means of removing all the doubtful points of divergent aeries from the ordinary branches of mathematics.
Next to the question of convergency or divergency, Collies that of continuity or discontinuity. We are not here speaking of continuity of value, but of form. A series is continuous when for all values of a it represents the same function of x. Thus or or .2° + XI + , is in all cases the development of (I whether it be convergent or divergent. Even those who reject divergent series altogether, though they would call this series, when x>1, a false or inadmissible development of (I would not, though they reject it, look upon it as possible to arise from any other function. But the series sin sin 20 sin 30 s, ( 1)"4-3 n0 0 2 3 " is discontinuous ; for certain values of 0 it represents one function, and for other values another. 'When 0 is any multiple of w [ANGLE] it is = 0 ; when 0 falls between 7r and + 7r, it is p ; when 0 falls between r and 3r, it is to 7r, &c. ; in fact, it stands for 10 nor, where in is to be so taken that to nor shall fall between 471- and + isr. Again, the series as 1 s3 (1+ a^ x) (1+ aa+lx) (a I) (x + 1) or (1a) (x + 1) according as a > 1 or < I : and when a = I, 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 But we mean that the arithmetical rattle 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 1 f 1 1 a-1 -11.iand consider as having a very great value, the second term will ho very small or very near to unity, according as a is > I or < 1. If we introduce the symbol a as representing en when a > 1, and 0 when a < 1, we have, on putting en for k, 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 tax can be developed into a +a,x+ ...' then ab +a,b,x+ + . .. can be developed as follows (D. C., p. 239). Let the last bo fx, and from 6, b &c. [DIFFERENCE] form Ab, &c. : then b = box + tab + ,a e'sb + + . . .
where cp'x, ex, &c., are the successive differential coefficients of 99.e. If b, b fie. be values of a rational and integral function of a, denoted by the preceding is not an infinite aeries, but a finite expression. We bass, not room for examples, and it is to be remembered that this is an article of reference. Particular classes of instances are b x tin 6 + b + = χ 0 + (1 " n-1 n 1 n 2 + 71 2 + r x n 1 (1 + I b + n + n + + b, + b, + . . = + Abx + + tho preceding is a case of more general theorem (D. C., p. 565) ham which the following may also be deduced : b vb. x V .
b 1),:e + = 1 + 4- (1 + mar + (1 + mar + ' V b = 6, + nib, = 6, + 2mb, + &c. By this theorem many divergent series may be converted into convergent ones, or the con vergency of convergent series may he increased.
sin o 2. Let r = (1 2 cos . x + tan = 1 x cos Then 6 + 6, cos 0 . x + b, cos 20. + b, cos 30. + , pbx cos cos + 20) + cos (28 + 30) + and b, sin 0 x + b, sin 20 . + 6, sin 30 . + b since + sin (0 + 73 + sin (20 + 399 , 3, Let 'x bo a rational and integral function of x; then 1 cps + (x + 1). a + (x + 2) . + . . . = cox +a a + qY''X (1 C15,x -r (1 2 1- (I 2 .3 + ' A, = a + + A, = a + 11a'+ + A, = a + + + 260 + &c.
A = a + + 302 + + + a5.
This must load to a finite expression for the aeries, and le frequently the shortest way of obtaining it.
4. Let re. oos e+ - 1 . sin and px=a + a,x + ....