4 
It may be observed that the summation of the series 
1 , 1 . . , 
+ : — 7 — — can be carried out to any 
A + w 
A + (n — 1 ) w 
required degree of accuracy by an application of one of Euler’s 
asymptotic series. Euler states (Inst. Calc. Diff., 1755, Pars 
Posterior, cap. VI) that 
1 + '.-i + 3 + • • • + — = y + log x -f ~~ 
x '> ' 
Z X 
P‘2 
2 x L 4 x 4 
'V + • • • where y = constant, and B lt B 2 , B . . . are Ber- 
6 x b 
noulli’s numbers, viz., B x = B 2 = ... An adaptation of 
the above formula gives 
1 
A 
I 
_1 
A -j- w 
_1 
A-\-{n — l)w 
w 
log 
A-\-(n- l)?y» 
A 
— w 
1 
2 (A 4- \ji- 1] w) 
B l w 
+ 
B x w 
2 (A~w) 2 (A-\-[n — \~\w) 2 2 (A-wj 
l 2 + 
r Ihe alternation of -f- and — signs makes it necessary to carry the 
series on the right to five terms, when the summation is usually ac- 
curate to several places of decimals. 
Relations between the powers of e. 
If, in Formula II, w is put = ] we get: — 
1 
A 
+ 
1 
2+1 
4- 
_i 
“T 
1 
A -f- 71— 1 
= 1 0 Q 
o 
■ ! 1 
+ n ( e 1/A 
»! 
or 
.( 
L+ 
A A - 
f 1 
-f ••• 
+ 
1 ) 
A + n- l/ = 
1 + n 
(e l / A 
1) = n. eA/A 
- n f 1, 
from 
which 
the 
foil 
ow 
ing approximate 
relations are ohtaii 
ed : — 
True Values. 
If 
n 
= 2, A 
= 
1 
A. 
B. 
f 
A. 
B. 
A 
= 
2 
e l + l = 
2e ■ 
1 
4 482 
4-437 
9 9 
A 
— 
3 
e\ + 1 = 
2 eJ - 
1 
2-301 
2-297 
cl+l = 
2e\ - 
1 
1 792 
1 791 
If 
n 
= 3, A 
— 
1 
e'+M = 
3e — 
2 
6255 
6*155 
»» 
A 
— 
2 
el+'+l = 
3ei - 
2 
2955 
2-946 
it 
A 
= 
3 
e,+i+ l 
5 
3 el - 
2 
2189 
2-187 
and so ad infinitum. 
