SERIES, summation of. We have be fore seen the method of determining the sums of quantities in arithmetical and geometrical progression. but when the terms increase or decrease, according to other laws, different artifices must be used to obtain general expressions for their sum.
The methods chiefly adopted, and which may be considered as belonging to algebra, are, 1. The method of subtrac tion. 2. The summation of recurring series by the scale of relation. 3. The differential method. 4. The method of increments. We shall content ourselves with an example or two in the first of these methods.
" The investigation of series, whose sums are known by subtraction." Ex. 1. Let 1 + 1 1 1 2 3 4 inf. =-- S, then 1 1 1 1 2 3 4 5 +, &c. in inf. 1 by subtraction, + + in inf.=1 ; or + = 1.
Ex. 2. Let 2 - - 4 3 &c. in inf. = S. Then 3+4 1 4 5 1 3 - &c. in inf. =-- S - by subtraction, 3 ' .
6 3+, &c. in inf. = 1 1 1 - 1 or + —+ — — &c. in inf.
1 . 3 2 . 4 3 . 5 4.6 3 = Ex. 3. Let 3 &c.
in inf. =--. S, 1 1 1 then + —+ .4-, &c. in inf. =-._ 2.3 3.4 4.5 • 2 2 by subtraction, E r 2 3 • 2.3.4 3.4.5 &c. in inf. aes , &c. in and 1 1.2.3 + 1 2.3.4 inf. 4' m 1 1 Er. 4. .L.,et - m+ 11 S, 1 1 then — + + • .% • M r in +2r m+Jr 1 1 Az — en + n r by subtraction, m-i-r in + 2r 1 + &c. (to n terms) -I- — =z ;
m+nr m hence, r+ &C.
in.ni + r . + 2r (to n terms) = 1 1 m ml-nr and 1 1 +, &c.
m + r .in + 2r 1 1(to is terms) = — mr 1 If n be increased without limit, - vanishes, and the sum of the series is 1 NS t'• If m = r z= 1, we have 2+ 21 3 + &c. (to n terms) = 1 — 1 1 +n Similar to the method of subtraction is the following, given by De Moivre.
"Assume a series, whose terms con verge to o, involving the powers of an in determinate quantity, call the sum of the series S, and 'multiply both sides of the equation by a binomial, trinomial, &c. which involves the powers ofx, and inva riable co-effIcients; then, if x he so assum ed, that the binomial, trinomial, &c. may vanish, and some of the first terms be transposed, the slim of the remaining se ries is equal to the terms so transposed." x x' s3 Let 1 4- - + &c. in inf=4, 2 3 4 Multiplying both sides by x-1, we have &c.
S.
x 5 T 4 3 or — 1 + x —4- x. —4-- +, &c =s-1 1.2 2.3 3.4 ; and if x v-- 1, 1 1 1 then,-4- +, tec.= 0; 11.2 -I- 2 3 1, 1 1 Or, 1-• — &c. in inf. 1.
1.2 2.3 3.4