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.