39 2 Mr. Vince’s new Method of 
Every feries of this kind may be refolved into the following 
•feiies + _ ^f+J * = + &c. for 
I,r + 1 r+ It 2r+I 2/ '+}» 3 '+l 3 r +1.4; + I 
if we reduce two terms of this feries into one, it will become 
lor— b 
+ “ 
1 y (l 4 * 2 ' — I . h 
‘ 1 — — - 4 - _ — 1 - _ — _ 
I . r + 1 . 2 r + i 2 r -f- I . 3>' + 1 . 4* 4- 1 <j.r 4- t 5/ 4- j . 6r 4- I 
where the denominators being the lame as in the given feries, 
and the numerators alio in arithmetic progreflion, we have 
only to take a and b fuch quantities th.at the refpedtive nu- 
merators may he alfo equal; all u me, therefore, 2 y a — b — m, 
2rm -f- n 
2 ' <7 4 - 4 — I . b 
&C. 
2 ra + 2 r - 1 . b = m + n ; therefore, b- n , a 
2 r 4 r~ 
fubftituted for a and b in lem. 2. gives 
m 4 - n tn 4 . 2 n 
W 
hich 
+ 
+ = 
1 . r 4 - i. 2 r 4 -i 2 r 4 - 1 . 3 r+i . 4 , 4- 1 4 r 4 - I . I . t>r 4 - i 
2 r m— r 4 - 1 * n v q _|_ 2r 4 * 1 - n — 2 rm 
2 r= 4, 3 
Let r — 1 , and we have 
-P &c. . . . 
m m + n m+zn 
-p *r 
1 - 2.3 3-4-5 5-6.7 
It 7/7 = i , n — 3 
&c. ... — m ~ n . S -p - 
77— 2W 
O’ 
+ 
I -. 2 . 
m— 1,72 = 0 , — - — + 
+ 
0 o-4-5 
I I I 
5 - 6.7 
+ 
•0 3-4-5 5 - 6.7 
Let r= 2 , and the feries becomes 
w m + n ;«4-2« „ 4 w— 3 « 
-p&C. . . . 7 - 2S; 
4 
-P 5cc. . . . = S — - . 
2 
1 . 
+ _^L±L + _JL!:fL +& c. ... ^ 
0 • 5 5 • 7 • 9 9 • 11 • *3 
1 ^ 
16 
s + 
57/-4OT 
o 9 
If m~ 1 , 7 ;= 1 , 
7/2=: 1, n — o, 
+ 
1 -3*5 5-7-9 
1 1 
+ 
+ &C. . . . = 4 - — ; 
l 6 22 
+ 
~P 
9 • 1 1 • 1 3 
T 1 c Si 
+ &C. ... = - - - 
4 
8 
1 • 3 • 5 5-7-9 9 • 1 * - * 3 
Let r t= 5 , and we fhall have 
m . m + n J.^ xS ^ 
9T OL.OT t /v p* 
I .6. I I T ’ 1 1 . 16. 21*21 .29. 31 
12 5 
5 00 
