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Mr. Vince on the Sums 
PROP. XVI. 
To find the value of aXjSx^xJx &c. ad infinitum , fuppofing 
the general term to he a rational function of x. 
Let nv be the general term, then refolve ^ into an infinite 
feries, and take the fluent on both fides ; then write 2, 3, 4, 
See. for x 9 and one fide will become the hyp. log. of the given 
feries, and the value of the other fide may be found from the 
tables. 
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Ex. 1 . To find the value of - x £ x — x &c. ad infinitum . 
3 8 15 y 
Here the general term is — — ; hence — = — 'fi— — - — 
x — 1 V A 3 — a- 
&c. ; hence the hyp. log. + — 4 + - I b + &c. 
Tx 2x 
v 5 v 7 
Write 2, 3, 4, See. for x, and we have the hyp. log. hyp. log. 
| +hyp. log. — + &c. = A + fC + §E + &c. = ,69314.7180574, 
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which is the hyp. log. 2; but hyp. log. - +hyp. log. | -f 
3 ' * “8 
hyp. log. ^ + &c. = hyp> log. | x x x &c. confequently 
- X | X — X &C. := 2. 
a 8 is 
Ex. 2. To find the value of - x — | x x Sec. ad infinitum. 
7 26 63 J 
Here the general term is — — ; hence - = — ~~ —^1 
7T A X X 
X J I I . I 
&c - ’ hence the h yp- lo §‘ + &c " 
Write 2, 3, 4, &c. for x 9 and we have hyp. log. — {-hyp. log. 
7 ^ ^ 
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