Proving $\left( 1-\frac{2}{n} \right )^{\frac {n\ln n}{4}}-\left( 1-\frac{1}{n} \right )^{\frac {2n\ln n}{4}}<0$

Question

As a part of a solution I'm writing I need to prove: $$\left( 1-\frac{2}{n} \right )^{\frac {n\ln n}{4}}-\left( 1-\frac{1}{n} \right )^{\frac {2n\ln n}{4}}<0$$ for large enough $n$. I checked in Wolfram-Alpha and it looks like it's true.. I've tried using:

$$1-x\leq e^{-x}$$ and

$$1-x\geq e^{-2x}$$

but I get $$n^{-\frac {1}{2}}-n^{-1}>0$$

Thanks.

Answer 1

The hint: $$\frac{\left(1-\frac{1}{n}\right)^2}{1-\frac{2}{n}}>1.$$ Thus, $$\left(\frac{\left(1-\frac{1}{n}\right)^2}{1-\frac{2}{n}}\right)^{\frac{n\ln{n}}{4}}>1.$$