invefiigating the Sums of infinite Series , 
393 
lim~ I, i, 
m ~ i, n~o, 
+ 8 fc. . . . r x S ■ ’ 1 
J 25 
I.6.1I 1 I.i6 .2I T 2I .26.31 
1 1 i Si 
, 4- : -f- 7 4* Sec. 1 • • — — — — • 
i.d.ii 11.16.21 2r.26.31 25 30 
5 00 » 
Cor. If 2r : r + 1 :: n : m, the fum of the feries can be ac- 
m 
Curately found, and will be equal to 
1 2r.r+l 
m — r+ij and then n — 2 r i confeqnently 
Let therefore 
— — == -f 
+ ■ 
— — - r ... ~ L • 
I . 2r 4- I 2f 4* I • 4' 4* I 4 ^ 4* 1 • 6 ^ 4- I *• 2 r * 
which is alfo known from other principles. 
PROP. II. 
m 
To find the fum of the infinite feries - -- c, 
r + 1 • 2 / -f- 1 . y 4- 1 
+ 
m + n 
+ 
m 4 - 2 n 
- 
- -f- &C. 
•/ 1 
11 r, 
3r4- I . + I • 5^4 - 1 5^4- I . 6r4- I . 7r4- I 
• • • • * * -J * , 
This feries refolves itfelf into 
c-\-b 
+ = l+lL=-& c; 
r-\- 1 . 2r 4- I 2r+i.y+i 3?- 4- I. 4/- 4- 1 
for by reduflioh, as before, it becomes 
2 cr — r 4 - I . b 2 cr+r — I . b ^ 2 WU\-$r— I . b - "> 
r 4- 1 . 2 r + 1 • y 4- i 3'- 4- 1 - 4r4- i . 5^4- 1 5^4- 1 . Or 4- 1 . 7' + 1 
where the denominators are the fame as in the given feries, and 
the numerators in arithmetic progreffion ; affume therefore 
2 cr — r + 1 . b — m 9 2cr + r—i . b — m 4- n, hence b — " r , 
sa TTLfiL which, fubftituted in cor. 2. lem. 2. give 
^ , i 
m + n m 4 - 2n 
— 4- — 1 — 1-— ====. 4- «==■ - 4 -&c. ... 
m 
r 4 -i . 2r4- I • 3 ^ 4- 1 1 ♦ 4? 4- 1 .. 5^4- 1 5 r+ 1 . 6 r 4 - i . 7/ 4- 
2 *'4- 1 ■ n — 2 rm ^ c ( 2 tut-* y-¥i*n t 2»'W — r — I . n m 
- x b 4 ~ ■ " 4 * 1 
3 
Vol. LXXII. 
F f f 
. r+ I 
Con 
