Thes whether the following series is convergent pointwise:
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{|f^n(x)|^k}$ , where $f:\Bbb R\to \Bbb R$ is monotone increasing function such that $f^n(x)\to -\infty$ as $n\to \infty$ for all $x\in \Bbb R$ and $k>1$ is a fixed constant.
Since $f$ is monotone increasing and $f^n(x)\to -\infty$ as $n\to \infty$ so we can get $f(x)<x$ for all $x\in \Bbb R$ and also $\displaystyle \lim_{n \to \infty}\frac{1}{|f^n(x)|^k}=0.$
But I'm unable to prove or disprove the convergence of the series.
Note: $f^n$ denotes the $n$-th composition of $f$. That is $f^n=f\circ f\circ \cdots \circ f$.
Can anyone help please ?
Edit: If it is not convergent then by imposing which extra condition on $f$, the series becomes convergent ?