Andre W. answered • 10/22/13

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Since f(x)=(ln(x))²/x is positive and continuous on [1,∞), the integral test is applicable to the series

∑

_{n=1}^{∞}(ln(n))²/n.Consider the improper integral

∫

_{1}^{ ∞}(ln(x))²/x dx = lim_{b→∞}∫_{1}^{b}(ln(x))²/x dx= lim

_{b→∞}[(ln(x))³/3]^{b}_{1}which diverges, so the series ∑

_{n=1}^{∞}(ln(n))²/n also diverges.