I was trying to determine if
$$\sum_{k = 1}^{\infty} \cos (\log k)$$
(with $k = 1, 2, \ldots$) converges. $\log$ is the natural logarithm.
I tried the ratio test,
$$\lim_{k \to \infty} \frac{\cos \left[ \log ( k + 1 ) \right]}{\cos \left[ \log ( k ) \right]} = 1$$
but it is not useful to determine the convergence.
According to Wolfram Alpha, it diverges. How to prove it?
And what about $\displaystyle \sum_{k = 1}^{\infty} \sin (\log k)$?
What I can empirically observe is that, regardless of how big $\log k$ can become, its cosine will always be a number $x$ such that $-1 \leq x \leq 1$. But I can't immediately see that this leads to an infinite sum.
Trivially, the term test is:
$$\lim_{k \to \infty} \cos (\log k)$$
and this limit is not defined, due to the cosine. Maybe this is sufficient to state that the series diverges.