RECURRING SERIES. 



mnltiplicrs by which it is contieftcd. Tntlic above we have law we have determined, and it not, we muJl iiicrcafc the 



ijfcd only two, /x and », whence ^ -f » is called the fcale ot terms in our fcale of relation ; for it may be oblerved, that 



relation, which is here of two terms ; and i minus the fcale we can never fail of determining them in confequence of 



of relation, as i — fi — », is called by the fame author the having aflumed too many terms, as we fhall, in that cafe, 



differential fcnU, which is always equal to the denominator of have one of ourrefults zero 



the fr:iftion from which the fcrics is produced. 

 If, in the above ferics, the terms had the relation 



y = '/ 



5 = IX.X. 

 t =: IJ.X . 



6 = lix . 

 &c. 



+ y x' .y + ( x'' 



+ v.r 

 &c. 



&c. 



then the fcale of relation would be |x + v + ?, which is of 

 three terms ; and the differential fcale, or the denominator of 

 the fraftion, is i — p — » — {. 



The following problems are naturally connefted with the 

 doftrine of recurring feries ; vix. 



1. Any recurring feries being propofed, to find the fcale 

 of relation, or the law of the feries, which is not always 

 obvious on infpeftion. 



2. To find the fuin of an infinite recurring feries, or the 

 fum of any number of its terms (n). 



3. To find a general expreliion for any indefmite term of 

 fuch a feries, as, for example, the ?;th term. 



We (hall confidereach of thefe problems under their fcpa- 

 rate heads. 



I. Tojind the fcale of relation in any propofed recurring feries. 



Let a, ;S, y, i, 1, 6, &c. be a recurring feries, of which it 

 is required to find the fcale of relation 



Alfume J = fi' y + »' ^ + &c. 



J = ^' ^ + v' y + &c. 

 9 = &c. &c. 



Where /S, 7, J, &c. are known, and fx, -i-, &c. unknown 

 quantities, whofe numeral values are required, and which are 

 readily found by the ufual method of elimination. Thus in 

 the above, ufing only f* and i, we have 



Lagrange has formed a different rule for afcertaining whe- 

 ther a given feries be recurring or not, which Mr. Bonny- 

 cadle has given at p. 323 of his Algebra; but as it does not 

 leem adapted for detedling the law of formation, we (hall 

 not infert it. 



IL To find the fum of any recurring feries, of which the 

 fcale of relation is known. 



) t e &c. 



X% £ X*, fx\ &C. 



- E(3 



f- = 



^<3' 



and 



H-' = 



26 jr* 

 49 x' 



-45*° 



4-^ 



-4»* 



= — I ;r' 



21 a" — 25.1'' 



when ^ = 2, and y ^ — 1. 



The fame method may obvioufly be employed in any other 

 cafe ; but in order to know whether or not we have affumed 

 a fcale of relation of a fufficient number of terms, we muit 

 repeat the fame operation upon three other terms, of which 

 one at lealt was not before employed ; and if both give the 

 fame values, we may be affured of the truth of our refults : or 

 we may otherwife, inilead of repeating the operation, exa^ 

 mine 



Let 



{:. 



y 



c r', 



be a recurring feries, of which the fcale of relation i» 



ft -)- V, fo that 



+ 0+7 + &C.) 



I — fj-x — tar 



which is a general expreflion when the fcale of relation is ot 

 two terms. 



When the fcale of relation is of three terms, fo that 



7 y — d (3 ^ — y^ 



Thefe values of ^', and ;', will obvioufly contain in them the 

 powers of the indeterminate quantity x, which being taken 

 out, we fhall have the required numeral values of n and v. 



Let it be required to afcertain the fcale of relation in the 

 feries 



a /S y 5 f 9 &c. 



I, ^x, sx\ 7 k', gx", II xS &c.. 



Here, by fubltituting the proper values of 0, y, 3, &c. in 

 the above equation, we have 



35 ^'-27^' _8;c' _ 



it is equally obrious that 



S = a + B + y + ^ + '■ + (!+ &c. or 

 + y + 



S = 



1.x (y + 3 + c -J- 



-M/9 + 7 + ^ + 

 ^x' {a. + + y + 



S = a-l-(3-{-y-l-fjA-(S - 



whence we have 



&c.) + 

 &c.) + 

 &c.), or 



/9) + V x^- (S - a) « *' + S 



s = 



+ J^ y - f..X {a. + 0) -^x" 

 I — a .V — V x'' — f ** 



which is the general expreliion for the fum, when the fcale 

 of relation is of three terms. 

 In the fame manner we have 



«4 g + y + ^— /•>»(« + g + y) — >x'(« + ^ —jx'^.a 



I — fix — vxj — gjc' — <rx* 



for a fcale of relation of four terms ; and fo on. 



, ..^ ^ ^ , .__ But as the terms a, /S, 7, ^, &c. contain m them certain 



the fcveral te'rmf, and fee whether they agree with the powers of m, we may reduce the above expreffions to fimpler 



forms 



