126 
ON THE SUMS OF CERTAIN SERIES 
[187 
This formula, which is easily deducible from that for the expansion 
powers of cos 6, is employed by M. Stern, Grelle, t. xx. [1840], in 
following theorem: If 
2 2.3 
continued to the first term that vanishes, then according as n is of the form 6 k + 3,. 
6& + 1, 6& or 6k + 2, 
8 = -, S = 0, S=- 1 , S = 2 , (9) 
n n n K ' 
of cos n6 in 
proving the 
(8) 
which is in fact immediately deduced from it by writing b = wa, w being one of the 
impossible cube roots of unity. Substituting the above values of x in the equation (4), 
( 1 + a ^- a + a ^.(» + i).{ 1+ ^^ + (P + »)ÇP 8 ^)P(P-l)^ | + ...| > 
(10) 
(1 + olY + (1 + a)-P = 2p j- + 
P g2 , (P + l)j>(p-l) a 4 , 1 
2 a +1 o q a. /«, » 
2.3.4 (a + 1) 2 1 -J 
(H) 
whence 
(1 + ay» + (1 + ay = (2p + 1) a [l + ( -£±+ P + ... 
+2p S + 
p a 2 
(7+1 + - 'T ’ = U su PP ose > (12) 
i.e 
A {-y (1 + ay = (-)^ +1 U or (1 + a)P = (~y 2 (~y +1 U, 
where A and 1 refer to the variable p. The summation is readily effected by means 
of the formulae 
2 (-)i ,+1 (2p + 1) (p + s + 1)...(p -s) = (~y (p + s+ 1)...(p -s- 1), 
2 (~) p+1 (p + s)...(p-s)2p = (-y(p + s)...(p -s-1), 
and we thence find 
(1 + ay = 11 + 
Pip- 1 ) a ~ . (P + 1)^(P — 1)(p — 2) a 4 ] 
1 + a 1.2.3.4 (1 + a) 2 + ‘"j 
1.2 
\P , (P + 1)P(P-1) g2 
+ a {i + 
1.2.3 
1 + a 
+ ••• . 
(13)