hi .e/d gating the Sums of inf nit e Series. 
_ 7 n-fez 
397 
i . 
r + I . 2 r + l . y f ; zr + i . y + I y + i . $r+ i 
+ &c. . . . r: 
2 r. r+ 1 
If r = I, 
-r 
+ • 
*3 
_ , . - 4 -&C. ... =— * ; 
1 • 2 • 3 • 4 3 • 4 • 5 • b 5 • 0 • 7 • 9 4 
r = 2 . 
+ 
r ' = 6 ? 
f- • 2 &c. ... — — * 
• Q • I I Q . 1 1 . 1 3 T r * 
17 
1 • 3 • 5 • 7 5 • 7 • 9 • 1 1 ' 9 • 1 1 • 1 3 1 5 ' 99 
5 + JI , *7 ■ __2 
I -7- 1 3* I 9 ' i 3- 1 9* 2 5-3 1 ' 2 5 * 3 x * 37 -43 336 
PROP. IV.. 
‘To find the Jum of the infinite f erics 
m 
m + n 
+ 
r + I . 2 r+ 1 , 3/ + 1.4-41 
m + 2 « 
+ 
3 r + I . 4r 4 \ . y + i . 0 r-f i 5r-f- i . 0r+ i . 7 r+ 1 . b> + I 
+ &C. 
This feries refolves itfelf into 
a a 4 b 
+ 
a-\-ib 
r + l . 2 r + 1 . y -\- 1 ar 4 1 . y -f 1 . 4/ + 1 y + 1 . y + 1 . 51-4- i 
for bv reduction it becomes = 
— & c.. 
ya — r + 1 . b 
ya+y—l ,b 
+ 
r + l . 2 r + I . 3r+ 1 . 4? + I 
4- &c. where 
ya+ 7r — I . £ 
3 r + I • 4^+1 • 5 r+ I . 6 r+ I 5^+ I • 6 r+i . yr+ I . br-f 1 
the denominators are the fame as in the given feries, and 
the numerators alfo in arithmetic progrefiion ; put therefore 
ya — ry 1 ,h~m^ ya + y- i . h — m + n, hence b = 
a = 4 rm which, fubftituted in cor. 2. prop. 2. give 
w 
4~ 
= 4“ &c. ... — 
r -\- 1 . ‘ if 4~ 1 ■ 3 r + 1 • 4 r 4~ 1 3 r + 1 • 4 r 4" 1 • 5 r + 1 • 9r4- 1 
T + 2 . n — 4^m c 2r;n — 4^4-1. n ( irm — r—z.ri yrm-zr-l.n 
1 x b d ^4 1- T „ rr-T + ~ 7 T=r 
1 2 . r 5 . r 4 ” I I2r 2 . r 4 - 1 . 2r+ I 
* Vide vi mot v re’s Mif Anal, pag, 134. 
Cor. 
