Difficult(?) convergence problem

[Atropos]

Homework Statement

Show that if $$\vartheta$$ is any constant not equal to 0 or a multiple of 2$$\pi$$, and if u$$_{0}$$, u$$_{1}$$, u$$_{2}$$ is a series that converges monotonically to 0, then the series $$\sum u_{n} cos(n\vartheta +a)$$ is also convergent, where a is an arbitrary constant.

Homework Equations

The Attempt at a Solution

I have attempted to show convergence via Cauchy's root test, Dirichlet's test, and Abel's test. All 3 of these attempts were unsucessful as one or more conditions required for the tests was not met.

 

[Stephen Tashi]

I have attempted to show convergence via Cauchy's root test, Dirichlet's test, and Abel's test. All 3 of these attempts were unsucessful as one or more conditions required for the tests was not met.

What are the objections to the Dirichlet test?

I'd think that $$| \sum_{i=1}^n (cos(n\vartheta)|$$ would be bounded since a run of positive terms is followed by a run of negative terms. Likewise for $$sin(n\vartheta)$$.

What's the longest run of positive terms that can happen? For $\vartheta > 0$ there is some smallest m so $M \vartheta > 2 \pi$ Intuitively, I'd think $2M$ would be plenty big to bound it.

 

[Atropos]

Wow, I'm an idiot.

I was so hung up on the cosine part of the sum that I completely forgot about the monotonic series u_n. I was only paying attention to the fact that cosine was sinusoidal ad therefore f_n>f_n+1>0 couldn't apply.

...but it does apply to u_n.

Thank you for pointing that out.