INCREMENTS. 



hy (Ttewing the application of the foregoing tlicory a few JUx. 3. — Find the fum of the fcries of fq\iare3 i' 

 esamplcs. 3- + 5- + 7' + &c. h\ 



Jpprtcalton of the Method of Increments. Here the general term of the feiics is (.v 4- 2) -, and there- 



fore the fum required will be expreffed by the integral of 

 {■V + 2)\ 



14. The fummation of feries, by the invcrfe method of 

 increments, is founded generally on this principle ; that if 



have any feries of quantities, as Now^ (.v + 2)' = y .r* -f /'4 ;r + ^4 ; and by 



b, e, d, e, &c. article 8 



which are deiived from each other according to fome known /• .v' .v* *• A .» 



law ; each term may be conlidered as the increment of the 

 fum of all thofe which precede it. Thus, 



l\^ 



6 



let a+i + c + d+e — z 4 ,/ '^ 



and (i + i + c + d + e+f=z*': 



2 ■^ X 



then it is obvious, that 2' — z = Az =/; and therefore, 4 /' i = i— ^ 



converfely, the integral of any one of thofe terms, con- ' -^ ■«■ 



(idered as an increment, will reprefent the fum of all the And fmce in this cafe A .v = 3, we hz\ 



preceding part of the feries. ;,.i ,.■> j ^ ^ ^t 



This being premifed, we may proceed to the folution of ^' (•» + -)' — 7 — t" ~g' + 



the following examples. ^ ^ ^ 



£x, I . - Required the fum of n terms of the natural feries 



3+4 + 5 



6 • 2+3-6 



by making .» = 



Here, by writing x in (lead of n, the term next in order 



will be a: + I, which being the increment of the feries, we {hall But here again, if the feries do not begin at unity, it will 



have /'( . + l) = the fum required. '/"^''''f % '^""'^'^'^i""' "''"ch will be found generally, thus : 



./ ^ ' ' ^ fuppofe the feries to commence at any term f ; and let x — p. 



Now /'(.r + i) = f '= +f^i andbyart. 8. then the above formula gives the fum of the fcries to the 



f X = — ^ — - ; and / I = —' term * = V + - + • whereas it ought to be p' ; the 

 "^ z Ax 2 -^ ^ ^623 ° ^ 



And fince in this cafe A .■ = I , we ha^•e ^^^^^^^.^^ ^^^^^^^^^ ^^^^^^^ _ .p_ P: p _p:\ 

 ■ •»■ _ ^^ + * _ " + " \6 ' 2 3 \/ 



/(«+0 = 



2 2' ^ ^ _ yL _ C. 4. £ \ = c ; and, therefore, the general expref. 



bv writing again n in (lead of x, which is the fum of n terms V » 2 3 , 



of the propofed feries, as is alfo evident from other confi- fion for any number of terms of this feries, between the limits 



derations. . , „ . r , /> and n, will be 



iJ^mari.— This example offers an eafyilluftrationot what "',«'' " P^ P^ P 



has been obferved at art. 9, of the corredion of an inte- Z^ 1'^ '^~ 6 '^ 1~ ^ ' 



gral. which is necelTary in many cafes, the fame as the cor- ^ J 



redion of a fluent is in the fluxional or dilferential cal- Qor. — This example will alfo furnifh the folution of the pro. 



cuius. Suppofe for example, that inllead of the preceding blem, when the roots of the fquares differ from each other by 



feries'beginning at unity, it had commenced from any other any condant quantity m; for it will only be neceffary to make 



term as 7 • the sreneral law of formation would have been , , r r i -n • /'/ n- 



term, as 7 , me gcucia. lav A* = m, and the fame formula wiU crive / f.v + m V = 



the fame, and the increment would ItiU nave had uie torm * > b j v t 1 



* + I ; and confcquently the integral, in the hrft inllance, f ^^ ^ 2 m f x + m^ J 1 

 would be reprefented as above ; viz. 



But here a correaion of the integral is neceffary, for 

 from the nature of the feries, when ;i- = 7. the fum ot the 

 feries is 7, this being the term at which the feries com- 

 mences ; whereas, without a correAion, we fliould have the 

 fum = 28 ; we mud, therefore, write 



r I. , ,\ _ •'' "^ •*' -\- c ; c being the corredlion, and 



ft nee when .t = 7, 



ll±l+,= 7, 



, . , . , c u f therefore the general formula for the fum beffiniiinK at any 



we find . = - 2 1 ; after which the fum of any """^ber of ^^^ ^ .jj ^^ h k 7 



terms of the propofed feries IS readily obtained. Thus ior r /,•„/, 



example, let « = 16, then the fum of the fer.es beginning ± + "_ + 1" ^ ±. +t _ 01/. , 



with the term 7 becomes 3 '» 2 6 3 " » 6 



16'-+ 16 •„] Ex. I. — Required the fum of the natural feries of cube* 

 i 21 = 115 as required. ^, _^ ^^i_^ ^, ^ ^, ^ ^^ ^,_ 



Vol, XIX. C Here 



