5Q6 philosophical transactions. [anno J 784. 



of any series whose general term is 



az m + £,m-l + &c 



z + e.z + e+l.z + e + 2 x &c. x Z+f.z +f+ 1 . z +/ + 2 . &c. x ; + g . z + g +T7&C. 

 X x-. 



And also from the sum of the serieses x -I &c. — 



e e + 1 ' e + 2 J f + \ 



X^ 1 X T^ 



-+• t-t— — &c. — -I ~— — &c. &c. can be deduced the sum of any 



/ + 2 g g + * g + 2 J 



series whose general term is 



<"" + ft'" 1 - + & c. x ^ z 



2*+ e.2z + e + l.&c. x 2j +/. 2: + / + 1 . &c. 2z +g. 2z +g + 1 . &c. 



The method of adding more terms of a given series together, as before taught, 

 may be applied to these and all other serieses. For example : let the given series 

 be 1 + %x + ix 2 -f \x 3 -\- &c. ; add two terms constantly together, and it be- 



comes 1 -f ±x + &c. = — (- -Z-— x 2 + -j^* + &c. = — - + 



4 + 3a ., 6 + 5a 4 c , .. , . • 2: + 2 + (2= + 1) 



— — - x 1 + — T-r~ x + & c - whence the general term is ■ v — ' 



— — — ■ a:". From the methods before given of addition, subtraction, and multi- 

 plication ; and the serieses found by this method, can be derived serieses, whose 

 sums are known. 



12. Suppose a given series axn -\- bx n ±s + cx^±2s _|_ dx't±3s _|_ &c. whose 

 sum p is either an algebraical, exponential, or fluential fluxion of x ; multiply 

 the equation p = ax n + bx n ±s _|_ cxm±.2s + d-x»— 3 * + &c. into o:±'-«, and 

 there results ax±r—np = a x± r .j. l, x ±r±s _|_ C ff±r±2* + &c; find the fluxion 



of this equation, and there follows - multiplied into the fluxion of the quantity 

 ( x ±>-np) = + rax±r-\ .L. (±r±s) bx±'±s-l -f (±r±2s) cx±r±2s—l _f. 

 &c. of which the general term is ( + r + zs) X /, where z denotes the distance 

 from the first term of the series, and t is the term in the given series, whose 

 distance from the first is z. In the same manner may be deduced the sum of a 

 series whose general term is t' X (i r i zs ) X ( + ?'+ (z+?w), or by repeated 

 operations t' X (ez 2 -\-Jz + g), where t' is a term of the given equation, whose 

 distance from the first term is z. And in general, from the sum of a given series, 

 whose fluxion can be found, and whose general term is t', can be deduced by con- 

 tinued multiplication, and finding the fluxion, the sum of a series or quantity, of 

 which the general term is At', where a is any function of the following kind 

 a'z'" -\- h'z m - ' -\- c'z'"" 1 + &c. in which z denotes the distance from the first term 

 of the series, and m a whole number. It is to be observed, that if the given 

 series converges in a ratio, which is at least equal to the ratio of the convergency 

 of some geometrical series, the resulting equation will always converge. But if 

 in a less ratio, then it will sometimes converge, sometimes not, according to the 

 ratio which the successive terms of the resulting series have to each other at an 

 infinite distance. 



