EXAMPLES: 
6 
Series. 
True Sum. 
Sum by 
Formula III. 
Approx. 
Error 
iu parts 
per 
100,000. 
1 
10* 4 
1 
ll 2 + 
1 
12 2 
0252089 
•0252193 
4- 41 
lb 2 f 
I 
n 2 + 
ad inf. 
105166 
•105263 
4 92 
1 
1 
4- 
1 
•90641 4 
•906455 
4- 5 
V10 
f a/11 
V12 
1 
fl'io 
1 
f I'll 
4 
1 
\/l2 
1-35059 
1 35063 
+ 3 
10 2 + 
ll 2 4 
12 2 
365- 
365 039 
4 11 
10 8 4 
ll 8 4- 
1 2* 
4059- 
4060*20 
4- 30 
VIO 
4 Vll 
+ 
V 12 
9-94300 
9 94284 
- 14 
fylO 
4 {'ll 
4 
^12 
6-66784 
6-66775 
- H 
V 100 
4- V101 
- ... 4- 
a/ 115 
165-8534 
1 
165*8516 
— i 
Derivatives of Formula II, 
Formulae for the approximate summation of series of the 
-f . . . can also be obtained by repeated 
form \ ^ . 
A m (A+w) 
differentiation of the empirical Formula II. E.g. : — 
I + A + rv + 
1 1 , 
= I()0* 
A 4- ( n ■ 1 ) w 'tv 
-1 
■* A 2 
1 
" A 2 
+ 
+ 
1 
(A + w) 2 
1 
+ 
1 
1 n.e u ' IA (—wlA 2 ) 
( A + \n-\ }w'y 2 w ' 1 +n(e w,A - 1 ) 
(A 4 w ) 2 
+ • • • + 
1 
(A+[n- 1] w)‘ 
1 
n.e 
w / A 
A 2 ' l+n(e WIA -Y) 
Formula IV 
