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

^{2}/n-
CREATE FREE ACCOUNT
- Access thousands of free resources from expert tutors
- Comment on posts and interact with the authors
- Ask questions and get free answers from tutors
- View videos, take interactive quizzes, and more!

- Become a Student
- Become a Student
- Sign In

Tutors, please sign in to answer this question.

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.

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.

Ralph T.

PhD Aero Engineer and 13 Year High School Physics and Math Tchr

Edmonds, WA

4.9
(78 ratings)

Raleigh E.

Raleigh-Physiology major from UW Seattle.

Bothell, WA

4.8
(90 ratings)

Annie H.

Experienced tutor who enjoys working with all ability levels

Bellevue, WA

5.0
(82 ratings)