Miscellanea 
137 
/ |2m \2 
.-. Multiply both sides by (^j^y^j 
/ j2»i \ 2 / |2m \2 / |2m \2 / |2m \2 
(0^ + (0= + (0^+-(0^ 
(since the number of terms is odd) 
^ 2 |j2^\ |2m \^~\m) J ' 
r Am Am |m\2 
The series 
/ in \^ „/ m.m-1 \- „/ m.m-l.m-2 
/m = r +2 = X +3 r -) + etc. 
^ ' \m + 1/ \m + 1 . m + 2/ \m + 1 . m + 2 . ?« + 3/ 
has the value \ m, a discovery I made when starting to table it. It is a particular case of a more 
general series which may be summed as follows : 
Consider the following identity : 
(m + m - 1)^ , 1 , fm - m - \\^ 
^ (m - 1) + m . I ) . 
4m \ m + m J 
, . , , (m - m - 2\2 
Multiply both sides by I ) . 
Then 
(m - m - 2)^ , 
[m - m - 2\2 (m - m - \ m - m - 
1 + m I . : ) . 
\m + m - 1/ \ m + m m + m - 1/ 
4m 
The process may be continued, thus 
(m - m - 2)^ 
(m + m - 2)2 , „, Aft - m - 2y „ 
^ - hn - 2) + m I I = (m - 2) 
4m \ 7ft + m / 
= (m -2) + (m - 1) 
+ 
4»i 
m-m-1 m - m - 
/ft + m - 1/ \ m + m m + m, - 1 
- m - 3 
Multiply both sides by { ^ — 
\m + m - 2 
Then 
(m - m - 3)2 , „s - m^^\2 , , ^ - m - 2 m - m - SV 
^ = {m - 2) I — I + {m - 1) I . . ) 
4m \fn ^ - 2/ \m + m- l m + m-2/ 
+ m 
m-m-1 m - m - 2 m - m - 
9ft + m jft + m - 1 m + m-2 
