invefii gating the Sums of infinite Series . 
3VJ 
*+•. 
1 ! 
fa-4 
5 
7 . 
9 
&c. — • 1 . 
60 
1 .2.3.4 
2 • 3 • 4 * 5 ‘ 3 
• 4 * 5 • & 
1 
!! 
w 
>4 
2 
3 
4 
Arr 1 - 
* • 3 -5 • 7 
3 • 5 • 7 • 9 ' 5 
.7.9. 11 
IS 
r== 3 > 
1 1 
* 7 - . 
2 3 
1.4.7.10 
4.7 . 10. 13 r 
7 . !0. !3 . 
OC L • • • « 
i(> 28 
PRO P: V. 
'To find the fum of the infinite feries — _ = - m • — — j- 
J r.r+i .ar+i .jr+i . 4 r+i 
m + n 
’+ 
m-\- in 
2 r-\- 1 . 3r + -I . 4 r + 1 . 5r + 1 . 6r-(- i 4 r + 1 .5* + I .6r + 1 . 7 r-\- 1 .8x 4 I 
This feries refolves itfelf into 
4 -&c. 
a-\-b 
a + lb 
r + 1 . 2r-j- X . 3r + 1 . 4 >'-j- 1 2r+ 1 . 3^+ I . 4 >'+ I • 5^4- 1 
by reduction this feries becomes 4 ' a ~ h 
+ 
I . r - {- I . 2> + 1 .37 + 1 
- &c. ; for 
4 r< 7 + br— 1 . b 
i . r + i. . ir + 1 . y f 1 . 47 - + 1 
4 Sec. 
\ra- f- I2r— 1 . b 
Ir 4- 1 -3>'+ I . 4 r 4 I .5^-+ 1 . 6r+ I 4 r+ 1 . 5 r-\- 1 .0r+ I .jr+ I 8r+ I 
where the numerators are in arithmetic progreffion, and the 
denominators the fame as in the given feries ; afl’ume therefore 
— b — m, 4 ra 4- 6r — i . b = m + n, hence b —~,a~ 
6 rm 4 r. 
' Z~ 5 
4 Va * * - 6r’ 24r- 
which values^ being fubftituted in cor. i. prop. 4. give 
m 
m-\-n 
4 ~^^* • . • — - 
1 . r 4- 1 . 2r 4- I . y + 1 . 4 r+ I lr+ 1 . 3^4 1 . 4 r 4 I . $>'+ 1 . br 4- 1 
Irm— ir + l . n 4 r-f- 1 . » — Irm 6 rnr+ Qn 2 rm-\-n 
’"'" s 72 .r + .r+i 2 4 r 3 . r -\- 1 . 2>‘4 !■ 
6 r $ * I 2 r 3 
Let r= i, and we have 
VI 
m + n m-\-ln 
; 4" — — + : , 0 ~ 7 .+^ C » 
i,.2.3.4.5^3.4*5* & *7 5.6. 7. 8.9 
2 m—eqi i^n—xbm 
X ^ 4 “ 
6 
72 
if 
