
1426 FINITE SUMS AND INFINITE SERIES
13.
n−1
X
k=1
k cos(2kx) =
n sin[(2n −1)x]
2 sin x
−
1 −cos(2nx)
4 sin
2
x
.
14.
n−1
X
k=1
a
k
sin(kx) =
a sin x − a
n
sin(nx) + a
n+1
sin[(n −1)x]
1 − 2a cos x + a
2
.
15.
n−1
X
k=0
a
k
cos(kx) =
1 −a cos x − a
n
cos(nx) + a
n+1
cos[(n −1)x]
1 −2a cos x + a
2
.
16.
n
X
k=0
C
k
n
sin(kx + a) = 2
n
cos
n
x
2
sin
nx
2
+ a
.
17.
n
X
k=0
C
k
n
cos(kx + a) = 2
n
cos
n
x
2
cos
nx
2
+ a
.
18.
n
X
k=0
(−1)
k
C
k
n
sin(kx + a) = (−2)
n
sin
n
x
2
sin
nx
2
+
πn
2
+ a
.
19.
n
X
k=0
(−1)
k
C
k
n
cos(kx + a) = (−2)
n
sin
n
x
2
cos
nx
2
+
πn
2
+ a
.
20.
n
X
k=1
2
k
sin
2
x
2
k
2
=
2
n
sin
2
x
2
n
2
− sin
2
x.
21.
n
X
k=0
1
2
k
tan
x
2
k
=
1
2
n
cot
x
2
n
− 2 cot(2x).
26.3 Infinite Numerical Series
26.3.1 Progressions
1.
∞
X
k=0
aq
k
=
a
1 −q
, |q| < 1.
2.
∞
X
k=0
(a + bk)q
k
=
a
1 −q
+
bq
(1 − q)