Convergence of $\sum\limits_{n=1}^\infty \frac{\cos(nx - n)}{n}$

Question

Trying to proof the convergence of (x in R):

$\sum\limits_{n=1}^\infty \frac{\cos(nx - n)}{n}$

I know it‘s trivially divergent for x = 1, and according to wolfram alpha it converges for x = 2 (or is WA wrong and it actually diverges?). Now that doesn‘t really give me a feel for it though. I tried Dirichlet‘s criteria, but was not able to get a good enough bound. The classic convergence tests don‘t seem very applicable (note that I may not use integral test).

Thanks in advance, ~whiterock

Answer 1

Since

$$\sin[(n + 1/2)z] - \sin [(n -1/2)z ]= 2 \sin\left(\frac{z}{2}\right) \cos nz,$$

we have

$$2 \sin \left(\frac{z}{2}\right) \sum_{n=1}^m \cos nz = \sum_{n=1}^m \left\{\sin[(n + 1/2)z] - \sin [(n -1/2)z]\right\} \\ = \sin[(m+1/2)z] - \sin \left(\frac{z}{2} \right).$$

That should help you find a "good enough bound" to apply the Dirichlet test for $z = x-1 \neq 2k \pi.$