The function is considered: $f: (0,\infty) \rightarrow \mathbb{R}, f(x)= \frac{\ln x}{x}$, how to prove that $(b_n)_{n\geq 1}$ has no limit?

Question

The function is considered: $$f: (0,\infty) \rightarrow \mathbb{R}, f(x)= \frac{\ln x}{x}$$

to be determined a string $(a_n)_{n \geq 1}$ so the string $(b_n)_{n \geq 1}, b_n= \frac{f^{(n)}(1)}{a_n}$ it has no limit

Answer 1

If $f^{(n)}(1)=0$ for all $n$ sufficiently large then the Taylor expansion of $f$ around $1$ shows that $f(x)$, and hence $\ln (x)$, is a polynomial in some interval around $1$ which is false. Now we use the following (to $c_n=f^{(n)}(1)$)

Lemma

If $(c_n)$ is not eventually $0$ then there exists a sequence $(a_n)$ such that $\lim \frac {c_n} {a_n}$ does not exist.

Proof: Take $a_n=c_n$ if $n$ is odd and $c_n \neq 0$, $a_n=-c_n$ if $n$ is even and $c_n \neq 0$, $a_n=1$ if $c_n=0$.