Evaluate $\sum_{n=1}^\infty \frac{1}{n}\frac{\ln{n}}{n-\ln{n}}$ [duplicate]

Problem_

Evaluate $$\sum_{n=1}^\infty\frac{1}{n}\left[\frac{\ln{n}}{n-\ln{n}}\right]$$

I separated the given fraction into two fractions: $$\frac{1}{n}\left[\frac{\ln{n}}{n-\ln{n}}\right]=\frac{1}{n-\ln{n}}-\frac{1}{n}$$

But, the terms are not erased by others. Then, how can I find out the values of each series? Actually, it is impossible to find out the sum of each series since both them diverge: $$\sum_{n=1}^\infty\frac{1}{n-\ln{n}}>\sum_{n=1}^\infty\frac{1}{n}>\sum_{n=1}^\infty2^n\frac{1}{2^n}=\sum_{n=1}^\infty1=(diverge)$$So the new idea should be established. Could you please suggest me other methods? Thanks.

[EDIT] I've also taken account of Riemann Sum, but I still don't know how to apply it.

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