Originally Posted by

**TwistedOne151** How would I determine and demonstrate whether or not the following series converges or diverges for a selected (arbitrary) real value for x:

$\displaystyle \sum_{n=0}^{\infty}\cos(2\pi\cdot(2^{n}x))$?

I've found clearly that it diverges for any x of the form $\displaystyle x=\frac{m}{2^k}$ with m and k being integers. I've also found that it diverges for $\displaystyle x=\frac{m}{3\cdot2^k}$. However, I don't know if it diverge for all x (as I suspect), or if there are real values of x for which the series converges. How do I find out conclusively?

--Kevin C.