Plane trigonometry, by S.L. Loney.

SUMMATION OF SERIES.

407

Hence, by equating real and imaginary parts, we have
C = 2 sin } cos, a)~.) 7r - a42
and S= 2 sin sin - a4
I 2 4
If a= 2nvr, clearly S= 0 and = oo.
EXAMPLES. LX.
Sum the series
1. sin a + g sin 2a + 2 sin 3a +...... ad inf.
2. cos a. cos a + os2 a cos 2a + cos3a cos 3a...... ad inf.
3. sina. sin a +sin2a sin 2a +sin3a sin 3a +...... ad inf., whereat+ a.
4. sin a. cos a + sin2 a. cos 2a + sin3 a. cos 3a +...... ad inf.,
where ak +.
2
5. sin a + c sin (a + ) + 2 sin (a + 23) +...... to n terms and ad inf.
6. 1 +c cosh a+c2 cosh 2a +......+ cn-1 cosh (n - 1) a.
7. c sinh a + 2 sinh 2a +............ ad inf.
8. 1 - 2 cos a +3 cos 2a - 4 cos 3a +...... to n terms.
9. 3 sin a+ 5 sin 2a + 7 sin 3a +...... to n terms.
10. When a-=, find what are the values of the series in Exs. 3
2i
and 4.
n (n- 1)
11. sina+nsin(a+p)+,.2 sin(a+2P)+...... to (n+l) terms,
1.2
n being a positive integer.
i 1.3
12. sin a + 2 sin 3a + 2- sin 5a +... ad inf.
22.4
13. cosn a - n cos-l1 a cosa+ - 2 cosn-a cos2a.. to(n+ 1) terms,
1. 2a...to(n+) terms,
n being a positive integer.
14. n sinain(1) 2a+n )(+ 2) sin 3a..... ad inf.
1.2 1.2.3