I
17
KINETIC ENERGY OF ELECTRICITY IN MOTION [1]
1
Since 4m²<1 for all the values concerned, we may expand
log (1-1/4m²), and thus obtain for the first part
-
n-1 1
m2 + उठे
n-1 1
+
mt
1
or, when we develop the sums by well-known formulæ,
=-
1
1
1
-#{const. -4(-1)+8(n-1)-24(n-1)
+
...
1
96(n-1)3
-
1)²
;+ ...}.
The constant is clearly log (2/π), for when n becomes
infinitely great the whole expression converges to n log (2/π).
If we develop the remaining terms in descending powers of n
and collect together like powers, we finally find the first part
to be
2\"
1
log +1+ +
П
8n
5
96n² +....
An analogous calculation may be performed for the second
part, which is
=
n-1
Σ 10g
log
1
2m2m
(2m-1)" (2m + 1)
=
1
n-11
m
=
n-1 1
+32 2
1
m³
n-1
Σ
log (1-1)
4m2
n-11
+ 128
and hence, by a calculation similar to the above,
= {0-577216
0·577216+ log (n − 1) +
∞ 1
+
+32 2
1m3
+18+
+1024
+...
M. P.
175
Σ
...
+...
1
+.
...
-
m5
+
-m
...
1
1
+ — +
C
2(n-1)
12(n-1)²
1
-... }
2(n − 1)² + .
}