real analysis - Does the series $sumlimits_{n=1}^{infty }left ( 1-frac{ln(n)}{n} right )^{2n}$ diverge?

could anyone help me figure out whether this infinite series

$$\sum_{n=1}^{\infty }\left ( 1-\frac{\ln(n)}{n} \right )^{2n}$$
diverges?

I've tried using Cauchy's and d'Alembert's limit tests but both gave the result 1. I've also tried the necessary condition for convergence, but

$$\lim_{n\rightarrow \infty }\left ( 1-\frac{\ln(n)}{n} \right )^{2n}=0$$

Answer

Use

$$\sum_{n=1}^\infty \Bigl( 1 - \frac{\log n}{n} \Bigr)^{2n} \leq \sum_{n=1}^\infty \biggl(\exp\Bigl( -\frac{\log n}{n} \Bigr)\biggr)^{2n} =\sum_{n=1}^\infty \exp( - 2\log n ) = \sum_{n=1}^\infty \frac{1}{n^2} < \infty.$$