I am interested in a computable series expansion of the following equation:
$f(n) = \cos(\log(n))$
Specifically, I am interested in real values of $n$ where $n>1$.
From basic series definitions of the two functions you have:
$\cos(x) = \sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m}}{(2m)!}$
$\log(n) = \sum_{k=1}^{\infty}\frac{1}{k}(\frac{n-1}{n})^k$
Can these two representations be defined in terms of one recursive relationship as follows:
$f(n) = \sum_{k=1}^{\infty}{g(n, k)}$
There seems to be a lot of information about logarithmic cosine functions but not the other way around for some reason.
According to Wolfram alpha, $$ \cos\log x = \sum_{n\geq0} \frac{1}{2} (-1+x)^n \left(\binom{-i}{n}+\binom{i}{n}\right). $$ Its radius of convergence is only 1, though. I believe that the $i$ in the above expression is $\sqrt{-1}$, and the binomial coefficient is defined using the $\Gamma$ function. So I'm not sure this expression is very helpful. If you only want the first few terms that you can ask alpha.