Converge/Diverge?

Question

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

Answer 1

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]^b₁

which diverges, so the series ∑ _n=1^∞  (ln(n))²/n also diverges.

Answer 2

It is clear that this series diverges because, apart from the first two terms, each term is greater than 1/n. 

The series  ∑ 1,∞ 1/n   diverges.