Series formed by reciprocal of fixed points of a function

Question

Let $S(f)=\{x:x>0,f(x)=x \}$, the series $\sum_{x \in S(f)}\frac{1}{x}$ converges for which function in the following?

(i) $\tan x$

(ii) $\tan x^2$

(iii) $\tan2x$

(iv) $\tan \sqrt x$

(v) $\sqrt{|\tan x|}$

By a quick estimation of roots for $f(x)=x$ one can rule out (i), (iii).

I have sketched a graph of each of the rest choices along with the line $y=x$ on each graph, I suspect (ii) to be the correct answer, but I am curious about a nicer way of showing it.

Answer 1

If $x=\tan(g(x))$ and $x$ is large then $\tan(g(x))$ is large so $g(x)$ is close to $(2n+1)(\pi/2)$ for some integer $n$ so $x$ is close to $g^{-1}((2n+1)(\pi/2))$. So you should be able to do a comparison test with $\sum_n1/g^{-1}((2n+1)(\pi/2))$.