If the integral test applies, use it to determine whether the series converges or diverges. ∑ 1, ∞ (ln(n))^{2}/n
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.