7 
And in the special case where n is infinite, 
+ 
1 , ] • f 1 e 
— 4- . . . ad nit. = — — • 
' a 2 
tv. A 
{A + w) 
wi A _ 
EXAMPLES : 
Series. 
True Sum. 
Sum by- 
Formula 
III. 
Error in 
parts per 
100,000. 
Sum by 
Formula 
IV. 
Error in 
parts per 
100,000. 
± + J: + J: 
10 2 ll 2 12 2 
•0252089 
0252193 
+ 41 
•0252032 
-23 
^ - + A + ... ad inf. 
10 2 ll 2 
•105166 
•105263 
+ 92 
•105083 
-79 
Other Series. 
By a modification of the foregoing methods approximate sum- 
mations can be found for geometrical progressions, and also such 
trigonometrical series as sinA+sin(A-[-- / M?)-f- ••• + sin (A-\-nw), 
but as these admit of exact summation the results are of no practical 
interest. 
My best thanks are due to Professor A. D. Ross for his assist- 
ance in putting this paper in a form suitable for publication. 
