Am i doing this right?/convergence of sequences

[miglo]

Homework Statement

$$a_n=(\frac{1}{n})^{\frac{1}{\ln{n}}}$$

Homework Equations

The Attempt at a Solution

$$\lim_{x\rightarrow \infty}(\frac{1}{n})^{\frac{1}{\ln{n}}}$$
$$y=(\frac{1}{n})^{\frac{1}{\ln{n}}}$$
$$\ln{y}=\frac{\ln{\frac{1}{n}}}{\ln{n}}$$
$$\ln{y}=\frac{\ln{1}-\ln{n}}{\ln{n}}$$
$$\ln{y}=\frac{\ln{1}}{\ln{n}}-\frac{\ln{n}}{\ln{n}}$$
$$\ln{y}=0-1$$
$$e^{\ln{y}}=e^{-1}$$
$$\lim_{x\rightarrow \infty}(\frac{1}{n})^{\frac{1}{\ln{n}}}=e^{-1}$$

the back of my book says the answer is indeed $e^{-1}$ but I am not sure if this is the way to go to for the solution or whether I am supposed to rewrite it as something similar to $\lim_{x\rightarrow \infty}(1+\frac{x}{n})^{n}$

 

[LCKurtz]

You steps are fine. That problem gives a constant sequence, just written in a non-obvious way.