SERIES. 



feries from that term, to which adding the fum of the firil 

 13 terms, we have, for the whole fum, 



13 firll terms - - = .674285961 



9 firll terms of the new feries = .018861219 



Whole fum 



= -693147180 



This is true to nine places of decimals, which, if we had 

 ufed the original feries, would have required the fummation 

 of at leall one hundred million of its terms. 



Hence the advantage of this transformation, which con- 

 Now, from what has been faid, it appears that 

 I I 



fids in our being able, by the fummation of a few of 

 the leading terms of the original feries, to give any de- 

 gree of convergency to our transformed feries, and thereby 

 to perform the fame upon a few terms, as would require the 

 labour of ages to effe6i upon the feries in its original form. 

 As another example, let the feries, 



I I I I I 



77 + ^ + y. + ^ + ^ + Sec. 



be propofed, in which the general term is — . 



+ 



x' X (x + 1) ' X [x + 1) [x + 2) 



and confequently the fum will be 

 I I 



+ 



+ 



*• (^ + I) (* + 2) (^ + 3) « (* + I) (* + 2H« + 3) (* + 4) 



+ &c. ; 



2j: (* + l) 3 X {x + l) [x -r 2) 



+ 



3 



+ &c. 



^x {x + l) {x + 2} (.V + 3) 

 indeterminate quantity enter ; all thofe which we haTC at 



in which fubftituting 13 for x, -viz, its 13th value, we find, 



by fumming 13 terms of the new feries, and adding tliat prefent confidered, having been wholly numerical. The 



fum — .079957427, to the (urn of the firit 12 terms of the formula for this purpofe is as follows. 



original feries, viz. 1.564976638, we Iiave 1.644934065 for If the terms of any feries be formed by writing any 



the whole approximate fuin, true to nine places ot decimals, number, differing by unity, for z in the quantity. 



Our limits will not allow of our entering farther upon c a h 



this method, and we (hall therefore conclude our illuftration 

 of it, by merely giving tiie author's formula for the fum. 



{1 



— + 



+ 



z (z 4- l) ' z (z -t- I) (z -)- 2) 

 raation of thofe feries, in which the fucceflive powers of an then the fum will be exprefled by 



b - Kx f— 2B« d— iQ.x 



\(r-l- 



X) 



+ 



(l-*)z(z4-l) il-j:)z(z + l)(z+ 2) ' (i-*)z(z4.l)(z+ 2)l.z + 3) 



+ &c.| 



where A, B, C, &c. reprefcnt the terms immediately pre- 

 ceding thofe in which they are found. 



This latter expreflion, like thofe in the preceding pro- 

 pofitions, will terminate when the feries is fummable : in 

 othir cafes, it will be itfclf an infinite feries, but fuch that 

 we may give to it any degree of convergency required. 



Simpfon^s Method of Series. — In I 743 Simpfon publifhed 

 his " Mathematical DifTcrtatioi-.s on a variety of Phyfical 

 and Analytical Suhjefts," and amongit other interefting re- 

 fearches in that work, there is one relating to the fummation 

 of feries, which is perhaps as general and complete as any 

 we Iiave yet noticed ; at Icail, if we except (with regard to 

 appr )xima'.ions) that of Stirling's, above explained. This 

 method confifts in deriving the fum of one feries from that 

 of another being given or known ; wliicli former fum is ex- 

 prefTed by a finite or infinite formula, according as the fuc- 

 ceflive differences of certain parts of its terms are of definite 

 or indefinite extent ; thus, if 



a" -I- ba"-' X f f a"-\v^ + da"-' x' + &c. 



be any power (n) of the binomial a + x, n being either in- 

 tegral, fraftional, pofitive, or negative ; and the terms of 

 it 'be refpeAively multiplied by any feries of quantities, 

 /) ?. r, r, &c. ; and we make q — p =z D', r — 2 q + p 

 = D", &c. viz. D', D", D'", &c. being the fird terms 

 of the fucceflive orders of differences, then will the fum of 



differences, D', D", D'", &c. become zero ; but, m other 

 caics, the new feries will alfo be infinite, the fame as that 

 from which it is derived. 



By giving to a, x, and n different values, and to the fe- 

 ries^, jr, r, s, &c. different laws, a great variety of parti- 

 cular cafes may be deduced, which our limits, however, 

 will not admit of detailing. 



Again, reprefenting (a 4- x)", as before, by 



a" + i a" + ' x + f a" + ' *•' + da" 



+ &c. 



if r be any pofitive number, and we make S = (j 4- *)"''■' 

 minus its firfl r terms, then will the fum of the feries 



+ 



ba"-' 



2 • .? • 4 • 



1.2.3. 

 4- &c. 



whether finite or infinite, be expreffed by 



S 



(r + i) 3. 4. 5. .(r 4- 2) 



q + ca"-'^ x' r + da" 



+ &c. 



a'p 4- ia' -• x 

 be cxprcircd by 



p(a + x)" + I)'bx(a + x)"' + D"fx' (a + x) 

 D"'dx' {a + x)"-' + &c. ; 



which formula will obvioufly be finite, if any order of the 

 Vol. XXXII. 



+ 



(« 4- i)(»4- 2) (n + 3)--- (» +'•)■■'' 

 From which general formula a great variety of particular 

 cafes may be drawn, according to the different values that 

 are given to a, x, and n. 



Again, let the fum of the feries 



a a' + bx*^" 4fA-'^"' 4 dx'^'" + &c. =A 



and the terms be refpcftively multiplied by the terms of the 

 arithmetical progrejfion r, r + n, r + 2 n, then will the 

 fum of the feries thence arifing (B), viz. 



rax* 4- (r + n)**'*" + (r + in) ex'*'- + &C. 



Oo be 



