82 PHILOSOPHICAL TRANSACTIONS. [anNO IJQI. 



Exam. 2. Let the general term be ^^^^ = i. + _L _|_ _i_ + &c.; hence 



if we write 2, 3, 4, &c. for or, we have ^ + i;5+]f^ + &c. = J-A + ^D+iG 

 + &c. = .219238483448. 

 By this prop, we may find the sum of any series whose general term is 



^ ; for this resolves itself into — — r-, ■ , , -, &c. &c., the 



sum of each of which series is found by this prop. Now the {s + l)th differ- 

 ences of the numerators of this general term are = O, and therefore it compre- 

 hends all series under such circumstances. For examjjle, let the given series be 



—• -1 — - -I- — - 4- TT-^. Here the 3d differences of the numerators = O; to find 



17 8'2 2'j7 o2o 



therefore the general expression for the numerator, assume ax^ -\- bx -\- c for it; 

 and, by writing 2, 3, 4, for x, we have 4a -)- 2Z) -j- c = 4, Qa -}- 3A -|- c = 

 13, l6a -{- Ab -\- c = lQ; hence a = 1, b = — I, c = — 2; and as the deno- 

 minator is manifestly x* -|- ] , the general term will be 



Or* ^ r ^ 9 ^X^ X 2 



— ^— - = ^— — — — — — — , each of which being made the general 



term of a series, their sum will be found to be respectively 1.077055849446, 



0.194173022145, and 0.156955159332; hence the sum of the given series is 



0.725927667969. 



If * be negative, the general term becomes 

 1^ _ _i_^ _ 1 , 1 _ - 



.T* X mx'±n mx''+' m^x'^' + ' ' m^xi'' + ^ 



Exam. 1. To find the sum of --,— , — r-v-: + . — &c. ad inf. Here 



1.2. J 2.3.4 3. 4. a 



the general term is , -r -. — — -r = — =z — 4- ~- -{- — 4- &:c. ; 



o (x — 1 ) X X X (JT + 1 ) J X (x' - 1 ) X' ' X* ' x' ^ ' 



hence, by writing 2, — 3, 4, — 5, &c. for x, we have the sum =^ b -\- d -^ y -\- 

 &c. = (by prop. 3) — - -|- 2 hyp. log. 2. 



If (^_i)xxix (x+i) ^^ ^''^ general term, it resolves itself into i + 1 -f i 

 + &c.; consequently the sum of — ^^ - ~~ + ^— _ &c. = - Z, - | 

 -I- 2 hyp. log. 2. In like manner the sum of ^-^ - — ^\--^ -f- 57^^,—- &c. 



=: — b — d — - +2 hyp. log. 2. Thus we may proceed as far as we please by 

 adding two powers to the middle term; and hence this remarkable property of 

 the serieses, that the difference of the sums of the serieses where the middle 

 term is x, x^, x\ &c. is b, d,f, &c. respectively. 



Exam. 1. In like manner if the general term be ; -r-^ — -. — -. — -— r, and we 



write 2, 3, 4, &c. for .r, we have -^ + 57^ + ~r^ + &c. = d -f f -|- 



