314 PHILOSOPHICAL TRANSACTIONS. [_ANNOl/82. 



are any number of deficient terms in the denominators, and where the last dif- 

 ferences of the numerators become equal to nothing; for if the factors in the 

 denominators be completed, and the numerators be multiplied by the same quan- 

 tities, their differences will still become equal to nothing. 

 Exam. To find the sum of the infinite series 



__L_ + -±— + —L- + _J°_ + »i__ 



1.3.4.0'^ 2 . 4 . 5 . 7 n 3 . 5 .6 . 8 ' 4 . 6 .7 .9 ~ 5 . 7 . 8 . 10 ~ 



This series, by completing the factors in the denominators and multiplying 

 the numerators by the same quantities, becomes 



•; — , , . + " — , , c , — ; + „,.,--„ + &c. m which case n = 1, 



1.2.3.4.3.0 2.3.4.5.0.7 3 . 4 . o . b . / . S 



m = 1, r = 5, and as the 5th differences are = O, 



Therefore a + b + 2e + 6d -J- 24<? = 10, a + 26 + 6c + 24d -f 120e = 54, 

 a + 3/f' + 12c -(- 60(/ + 360e = 1 68, a + 4£ + 20c + 1 20d + 840e = 400, 

 a + 5b + 30c + 210d 4- l6sOe = 810; whence a = O, b = O, c = — 1, 

 d = 0, e = -i-; consequently the sum of the given series is = — 

 _L_ -i ' — 17 



3.4.5.3 ' 5 .2 — 180' 



By a method similar to that made use of in this proposition may any number 

 of factors be taken from the denominators of those series delivered in part the 

 1st, and also from a great variety of others; but as the examples here given 

 must be sufficient to point out the method of proceeding in all other cases, we 

 may proceed to the 3d part. 



Part 3. — The sum of every converging infinite series, whose terms ultimately 

 become equal to nothing, may always be exhibited by the sum of another series 

 formed by collecting 2 or more terms of the former series into one. This is not 

 true however where the terms of the infinite series continually diverge, or con- 

 verge to any assignable quantity, and are affected with the signs -}-, — , alter- 

 nately: for instance, the series 4- — ■§- + -§- — 1+ &c. if we collect 2 terms into 

 one, beginning at the 1 st term, will become — — — — — — ^— r + & c . Jf vve 



begin at the 2d term, it becomes - — ■ + — — 4- — -, -f &c; neither of which 



° 1 . Z 3 . 4 o . 



gives the sum of the assumed series; but in this, and every other case of the 

 like nature, a correction will be necessary: to determine the value of which, and 

 whence the necessity of it arises, is the subject of this 3d part. 



Lemma. — If r be any quantity whatever, then will — = 1 l. 



Jkc. ad infinitum. 



For — = = (by common division) 1 4- &c. ad infi- 



2r r + r v J ' r r r r 



nitum. 



Cor. 1. Hence — — = — 4 1 &c. ad infinitum. 



2r t r >' r r 



