the Value of a slowly converging Series. 187 
of its terms, much more swiftly than the powers of ~o> anc h m 
the first seven terms, much more swiftly than the powers of 
-h- The value of the series | ^ • Sf- tV • 
&c. is therefore = the series — *—-4 — — . JLl _i 1 — 
1 9 19 9 AO 19] '9. 10. 11 
2 -3 9 
9 _ 
19 
+ 
+ & c - the first seven terms of which con- 
verge above fourteen times as swiftly as the other ; or, in other 
words, the first seven terms of it will give a result much nearer 
the truth than a hundred terms of the other. And if, instead of 
the first eight terms of the proposed series, the first twenty- 
four terms were computed, as they stand, and then the value of 
the series £ - | | &c. by its equivalent,^ ~f+ 
+ 
25.26 ' 25.26.27 (1+x) 3 25.26.27.28 
2 -3 
rapid decrease of the coefficients 
25’ 25.26’ 25.26.27* 
-f- &c. the 
&c. com- 
pounded with the decrease of the powers of (in the pre- 
sent case = the powers of T 9 ^,) produces such a very swiftly 
converging series, that eight terms of it will give the result true 
to eleven places of decimals. 
It may be further remarked, for the sake of my less expe- 
rienced readers, (for whose information this article is chiefly 
intended,) that the second term of the last series is produced 
by multiplying the first by ^ ; the third, by multiplying 
the second by — • the fourth, by multiplying the third by 
28 * TTx » anc * so on - therefore, the first term be called P, 
the second Q, the third R, the fourth S, &c. we shall have. 
Bb 2 
