Mr. Vince on the Sums 
304 
found by this propofition. Now the j-p-ith differences of the 
numerators of this general term are = o, and therefore it com- 
prehends all feries under fuch circumftances. For example, let 
the given feries be —■ + — +■— + ~ . Here the third dif- 
0 17 82 257 626 
ferences of the numerators = o ; to find therefore the general 
expreffion for the numerator, affume ax r -\-bx-\-c for it ; and, 
by writing 2, 3, 4, for#, we have = 4? 9 aJ r^ J r 
— 1 3, 1 = 26 ; hence a = 2, - 1, c — - 2 ; 
and as the denominator is manifeftly a; -}- 1 ? ^ le general term 
will be — 7 * ~ -7— — ■* - , each of which being 
made the general term of a feries, their fum will be found to 
be refpe&ively 1,077055849446, 0,1941 73022145, and 
°,i 56955 159332 ; hence the fum of the given feries is 
0,725927667969. 
If s be negative, the general term becomes — h— - 
x x mx ±: n mx 
m x 
zr-\-s 
mx 3 1 
&C- 
Ex. 1 e To find the fum of — - 1 — - &c. 
1.2.3 i-3-4 3-4*5 
ad inf. Here the general term is = — - — ===== — - l - • = 
J x — I I xxx 1 — I 
2- + -T + 4 5 hence, by writing 2, — 3, 4, - 5, &c. for 
we have the iwm — &c. = (by Prop. 3.) — . 
4 . JL 2 hyp. log. 2. 
If = be the general term it refolves itfelf into 
i + ^ + ^+ &c ’5 confequently the fum of 
+ 
