Convergence of a sequence with natural logs

Question

Let $c>1$ be an integer constant.

Edit: I am wondering if the sequence $$ cn\left[\frac{1}{\ln\left(n\right)} - \frac{1}{\ln\left(c n\right)}\right] \qquad \mbox{converges as}\quad n \to \infty. $$ My first guess is that it should go to $0$, but simulation results show otherwise.

Answer 1

Response to the updated question....

Notice that

\begin{align} cn \left(\frac 1 {\log n} - \frac 1 {\log cn}\right) &= \frac{cn \log c}{(\log n)(\log n + \log c)}. \end{align}

Now $\log(n)^2$ is far smaller than $n$ once $n$ is large enough, so this tends to infinity. This almost grows linearly in $n$.