Plane trigonometry, by S.L. Loney.

SUMMATION OF SERIES.

409

and S = 2 nsin (c sin ) [ecos 0 - e-ccos ]
= sin (c sin 0) sinh (c cos 0).
Aliter. From (3) we have
1 1
C + Si = - e( sin 8-i cos O) i + ~ e-( sin 0-ic cos 0) i
2 2
= cos (c sin 0 - ic cos 0) (Art. 307)
= [cos (c sin 8) cos (ic cos 8) + sin (csin 0) sin(iccos 0)]
= [cos (csin 0)cosh(ccos 0)+isin(c sin ) sinh (c cos 0)]
(Art. 314).
Hence C and S as before.
355. Ex. Sum the two series
c2 C3
c sin a + sin 2a + sin 3a +...... ad inf.,
2 3
c2 c3
and c cos a + - cos 2a + cos 3a +...... ad inf.,
2 3
where c is numerically not greater than unity.
Let S and C stand for these two series; then, as
before, we have
C + Si = c(cos a + i sin a) + 2 (cos 2a + i sin 2a) +......
2 3
=-log [1 - cea] (by Art. 336)........... (2)
= - log [1 - c cos a - ic sin a] (Art. 308).
Let 1 - c cos a = r cos 0, and - c sin a = r sin 0,
so that
- ------- ~1 - c cos a
r= + - -2c cos a + c, cos 0= -,
r