5Q4 PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



rz+(n— 2)r+e' "•" c - + a \ 77+7' >' z + r+f ' ' ' r*~+~jm~Z)r~+f' ~^~ 



( r 7+T+7- • ■ + 7T+\^zrz)7T. 7^ + &c " 



2. If the serieses are 1- _ — & c . and -.— -. 1- — 



e e + r ' e + 2r J J + r ' f + 2r 



&c. ; then from the sum of these two serieses can be collected, by the prin- 

 ciples above given, the sum of any series whose general term is 



av» + iiz m -^ + yz'"-2 + &C. 

 2rz + c.2rz + r + e.2rz + 1r + e ... 2rz + («+]) + e + 2rz +f.2rz + r+f.2rz + 2r+f...2rz + (m— l) r+f 



The same principle may be applied to find the sum of any series of the above- 

 mentioned sort, in whose denominator are contained other factors, rz -J- g, rz -f- 

 g -\- r, &c. &c. , or Irz -(- g } 2rz -\- g -\- r, 2rz -j- g + 1r, &c. Like pro- 

 positions may be deduced from serieses, in which r and r, &c. and the factors 

 rz -\- e and rz + g, &c. denote different quantities. 



10. An apparently more general method may be given from assuming a series 

 or serieses as before ; and adding every 2, 3, 4, &c. (?i) successive terms together, 

 for terms of a new series beginning from the 1st, 2d, 3d, &c. ?i ,h term ; and in 

 general adding together 2, 3, &c. n successive general terms ; and in their sum 

 writing for z the distance from the first term of the series 2z -f- «, 3z -}- a, &c. 

 hz -f- a ; there will result the general term of a series not to be found from the 

 above-mentioned addition. 



Ex. Let the series assumed be-^-|-4--(--i- + i-r- &c. in infinitum, of which 



the general term beginning from the first is ; add 3 successive general 



i . l . 1 3: 2 + \2z + n . ,,• , c 



terms = — •— ; in this term tor z write 3z, 



z+\~z+2~z + 3 Z + 1.Z + 2.Z + 3' ' 



27^ + 36^ + 11 



and there results - — ; -^-- . In the same manner, if the beginning; is 



3; + 1 . 3- + 2 . 3: + 3 to ° 



instituted from the 2d or 3d term of the given series, the terms resulting will 



, 3z 2 +18z + 26 , 3s" + 24J + 47 T ., . ,- -. „ , 



be — -—— —-— ; and . In these terms for z write 3z, and 



s + 2.s+3.= +4 s + 3.s + 4.2 + 5 



there result s^ + ^+ JL and —- 27 -!-±I£i±iZ_ . 



3= + 2 .3; + 3.3s+4 3s + 3.3s + 4.3s + 5 



If the terms of the given series are alternately affirmative and negative, the 

 terms of the resulting series will be alternately affirmative and negative, if n be an 

 odd number ; otherwise its terms will be all affirmative. The sum of this series 

 will be finite or infinite, as the sum of the series 1 + 4. -|- -J- + + -f- &c. is finite 

 or infinite ; but from it, by the preceding method of addition or subtraction of 

 Mr. Bernoulli's, or a like method applied to more serieses, may be found the sums 

 of different finite serieses. It may be observed, that from Mr. Bernoulli's addi- 

 tion or subtraction can never be deduced the serieses which arise from this me- 

 thod ; for, by his method, the denominator can never have any factors but what 

 are contained in the denominators of the given series, viz. (in the series + + } + 



