3 
Putting n = 1, 
1 
A 
w 
. log 
V 7 
A - 1 
wf A 
er 
w 
A — \ w 
1 . 
-k 'W 
Substituting this approximate identity in the equation above we ob- 
tain the following formula which holds even when n is not large : — 
A 
+ 
1 
A + w 
• • ■ + 
A -f- (n — 1) w 
- . log fl n U u, / A — 1)} Formula II. 
w ' v ' > 
EXAMPLES : 
Serie?. 
True Sum. 
! Formula I. 
Approx. 
Error in 
parts per 
100,000 
' Formula II 
Approx. 
Error in 
parts per 
100,000 
1 
+ 
1 
•83333 
•83939 
+ 727 
•83180 
-184 
1 
2 
+ 
+ 
l 
10 
1-92897 
1-94429 
+ 795 
1-92257 
-332 
1 
2 
+ 
. . . 
+ 
1 
25 
281596 
2-83284 
+ 601 
280755 
-299 
1 
10 
+ 
I 
II 
+ 
1 
12 
•27424 
•2 428 
+ 15 
•27423 
-4 
1 
10 
4- 
+ 
1 
20 
•76877 
•76897 
+ 26 
•76866 
-14 
1 
10 
-t- 
. ( • 
+ 
1 
25 
•98699 
•9872 
+ 26 
•98683 
-16 
1 
100 
+ 
l 
101 
•0199010 
•0199010 
p 
•0199010 
p 
1 
ioo 
+ 
+ 
1 
115 
•14911 
•14911 
? 
■14911 
p 
i 
100 
+ 
+ 
1 
150 
•41380 
•41380 
? 
•41380 
? 
