88 
SERIES 
6. As an application of 5 we see the alternating series 
-6 = f + tg • * * 
is convergent. For as the A series we may take A = 1 —1 + 1 — 
1 + ••• as I A n \ < 1. 
84. Trigonometric Series. 
Series of this type are 
C = a Q + ¿q cos x + a 2 cos 2 x + a 3 cos 3 x + ••• 
S = a x sin x + a 2 sin 2a: + a 3 sin 3 x + ••• 
As we see, they are special cases of Abel’s series. Special cases 
of the series 1), 2) are 
r = | + cos x + cos 2 x + cos 3 x + • • • 
2 = sin x + sin 2 x + sin 3 x + ••• 
It is easy to find the sums 2 n as follows. We have 
(1 
(2 
(3 
(4 
2 sin mx sin 1 x 
2 m — 1 2 m + 1 
COS X — COS' x. 
2 2 
Letting m — 1, 2, ••• n and adding, we get 
o * 1 x* i 2 n -j~ 1 
2 sin A x ■ 2„ = cos A £ — cos —+— 
(5 
Keeping x fixed and letting n = oo, we see 2 n oscillates between 
fixed limits when x4=- 0, ± 2 7r, ••• 
Thus 2 is divergent except when x= 0, ± 7r, ••• 
Similarly we find when x=fc 2 mir, 
-p _ sin (n — ^A x 
n cî • i 
A sin £ X 
(6 
For 
Hence for such values oscillates between fixed limits 
the values x = 2 mir the equation 3) shows that T re = + oo. 
From the theorems 4, 5 we have at once now 
If 2 | a n+l — a n | converges and a n = 0, and hence in particular if 
••• = 0, the series 1) converges for every x, and 2) converges 
for x=P‘l mir. 
If in 3) we replace x by x + ir, it goes over into 
A = \■ — cos x + cos 2 x — cos 3 x + . • • 
a