The problem of numerical evaluation of the classical trigonometric series
\[
S
ν
(
α
)
=
∑
k
=
0
∞
sin
(
2
k
+
1
)
α
(
2
k
+
1
)
ν
and
C
ν
(
α
)
=
∑
k
=
0
∞
cos
(
2
k
+
1
)
α
(
2
k
+
1
)
ν
,
{S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},}
\]
where
ν
>
1
\nu > 1
in the case of
S
2
n
(
α
)
{S_{2n}}(\alpha )
and
C
2
n
+
1
(
α
)
{C_{2n + 1}}(\alpha )
with
n
=
1
,
2
,
3
,
…
n = 1,2,3, \ldots
has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when
α
\alpha
is equal to a rational multiple of
2
π
2\pi
, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving
C
ν
(
α
)
{C_\nu }(\alpha )
and
S
ν
(
α
)
{S_\nu }(\alpha )
in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established.