Dr. Waring t . 
84 
; affume, by the preceding method for the fum of 
, of which the fucceffive 
1 . 2 . 3 . . a 
the feries the quantity — 
term is 
1.2.3 
1 
2 . 3. . Z-l 
and their difference 
1.2 • • Z — l 
z 
— the given term. 
I . 2 . . 2-2X ss * 2 * * * & * * * 10 , 
2. Let the term be Ne* and e lefs than 1, which is the term 
of a geometrical feries ; then will the fum of the infinite feries 
1 N 
be — X e z , beginning from the term whofe diftance from the 
firft i6 z ; for the difference between the two fucceffive fums = 
— ^ (e* — r K+I ) = Ne a the given term. 
2. Let the general term be ( z d- » + 1 ~ e _ x ^+1 . a ff U me 
for the fum of the feries the fubfequent quantity (% -j- i . z+ 2 • 
% + 3 . . z + n)— x x e* x + and by the 
preceding method the co-efficients a, / 3 , 7, &c. may be found : 
the fum is known to be =: -L- x £* +I + X x 
Z+I Z + 2 Z+3 
• • which can eafily be reduced to the preceding 
formula. 
*p/ __ »p p Q/ Q p/ 
If the general term be — -7 or ; where T and T', 
1x1 
P and P', Qjind Q^, are fucceffive terms ; then will the fums 
1 P 
of the feriefes be — or ~ properly correded. 
10. If the function expreffing the general term contain in the 
denominator a fador or fadors, which have no fucceffive one ; 
reduce the fador or fadors into an infinite feries proceeding 
according to the dimenfions of %, and thence, by the method 
before given, find the fum of the feries. The fame method 
4 may 
