investigate the convergence of $$\sum\_{n\ge 2}\ln\frac{n^2}{n^2-1}$$

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I solved this problem:

the series converges because 
$$\int\_1^{\infty}\ln\frac{x^2}{x^2-1}dx=2\ln 2<\infty$$
but the integral needs lines of computations, I would like to ask if somebody sees easier/faster convergence proof.