How to prove that this series converges uniformly?

Question

I have a series $$ -\frac{\pi}{12} + \sum_{k=1}^\infty \frac{\left(3k\pi^2-16\right)\sin{\frac{k\pi}{2}} + 8\pi\cos{\frac{k\pi}{2}}}{\pi^2k^3}\cos{kt} $$ And I have to use Weierstass test to prove that the series converges uniformly. The problem is that the sines and cosines are really making it hard to simplify the absolute value of the above and compare it to a known convergent series. Could you help me with that?

Answer 1

The series converges absolutely and uniformly, independently of $t$. We can bound the numerator above by $3k\pi^2 + 16 + 8\pi$, which is in general less than $C k$ for a constant $C$ and entirely independent of $t$.

Then the sum looks like $$ \sum_k \frac{C}{k^2},$$ which converges absolutely. This gives absolute and uniform convergence by the Weierstrass M test.