I am trying to show that if $\theta$ is not equal to $0$ or a multiple of $2\pi$, and if $u_0,u_1,u_2,...$ be a sequence such that $u_n\rightarrow 0$ steadily, then the series $\sum u_{n}\cos(n\theta +a) $ is convergent and also if the limit of $u_{n}$ is not zero, but $u_{n}$ is still monotonic, the sum of the series is oscillatory if $\theta\over\pi$ is rational, but that, if $\theta\over\pi$ is irrational, the sum may have any value between certain bounds whose difference is $\lim _{n\rightarrow \infty }u_{n}cosec\dfrac {\theta } {2}$.
Solution Attempt Based on the upper bound of the partial sums are bounded as shown by Julián Aguirre below and based on G.H.Hardy's test for uniform convergence which states that if, in a given domain, $\left| \sum _{n=1}^{p}a_{n}\left( z\right) \right| \leq k$ where $a_{n}\left(z\right)$ is real and $k$ is finite and independent of p and z, and if $f_{n}\left( z\right) \geq f_{n+1}\left( z\right) $ and $f_{n}\left( z\right)\rightarrow 0$ uniformly as $n\rightarrow \infty$, then $\sum _{n=1}^{\infty }a_{n}\left( z\right) f_{n}\left( z\right) $ converges uniformly.
Since $u_n\rightarrow 0$ and the sequence $u_0,u_1,u_2,...$ is monotonically decreasing. We also observed that the partial sum is bounded so the series converges uniformly. Hence the series must converge.
I am unsure how to attack the second part of the problem, to be more specific how the cos function varies when the parameter part is irrational. Any help would be much appreciated.