VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 5Q1 



finite terms of the quantity z. In like manner, if the dimensions of z in the 

 numerator are less than its dimensions in the denominator by urtity, then the in- 

 tegral of the general term cannot be expressed by a finite algebraical function of 

 z. If the number of terms to be added be infinite, it is well known that the 

 sum in this case will be infinite. 



It may be observed, that in finding the sum of a series, whose general term 

 is given, all common divisors of the numerator and denominator must be re- 

 jected, otherwise serieses may appear difficult to be summed, which are very easy : 



for example, let the series be M ^ 5 + 4 . 5 .g, 7 . 8 + f^^JoTn 



+ &c - = J (iTsTTTs + rrr?Ti + tTtAoTiT + &c -)' whose s eneral term is 



2 + 1 



; and by assuming, as is before taught, (3z + l) 



3; + 1 . 3: + 4 X 32 + 2 . 3z + 5 



X (3z + 2) - ' X « for the sum sought ; and finding its general term (3z +1)-' 



X (3z + 4)- 1 X (3z + 2)-' X (3z + 5) _1 X 18 (z + 1) X a, which equating 



to the general term given, there results 18« = 1, and the sum sought 



-± x - . 



18 3z + 1 .3:+ 2 



Ex. 2. Let the series be —-±—- + j—^ + 9 . 10 . t f , 12 . 13 + 



la. u AT. i6. 17 + &c - = h, <rh + nr§ + <nT3 + r 3 -7i7 + &c -)> of which 



the general term is — X : — tt ; an d consequently the sum deduced is 



o ox 4z + 1 . 4: 4- 5 ' J 



2-t 4z + 1 . 4; + 5 



1 X - X — — 



24 X 4 A 4c + 1* 



These are serieses given by Mr. De Moivre, and esteemed by Dr. Taylor 

 altioris indaginis. Some other writers have made some serieses to appear more 

 difficult to be summed, by not reducing them to their lowest terms. 



7. Having given the principles of a general method of finding the sum of a 

 series, when its general term can be expressed by algebraical, and not exponen- 

 tial, functions of z, the distance from the first term of the series ; it remains to 

 perform the same when exponentials are included. 



1 . Let s the sum be any algebraical function of z multiplied into e z = x z ; 

 then will the general term be se K — es'e x = (s — es') e* ; whence, from the ge- 

 neral term Te z being given, assume quantities in the same manner (with the 

 same denominator, &c.) as when no exponential was involved, which multiplied 

 into e z , suppose to be the sum ; from the sum find its general term, and equate 

 it to the given one by equating their correspondent co-efficients, and it is done. 



~ 4- 2 



Ex. Let the general term be o~ + ~ i ~'o^~7r 3 X eS+I : assume for the sum sought 



— !Jri x &e+I ' whence the g eneral term is (5747-7— WT^ e *~ l = 



