Boundedness of derivative of logarithm functions.

Question

We know that $\ln^{(k)}(x)$ = $\frac{(-1)^{k-1}(k-1)!}{x^{k}}$. The questions is: are all the derivative of logarithm functions bounded? And bounded by what? What confues me is that what if when $k \rightarrow \infty$.

Thanks.

Answer 1

$\sum \frac {x^{k-1}} {(k-1)!}$ is a convergent series for each $x >0$ so the general term $\frac {x^{k-1}} {(k-1)!}$ tends to $0$. It follows that $|(-1)^{k-1}\frac {(k-1)!}{x^{k-1}}| \to \infty$.