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.

Melissa R.

Penn State Chemical Engineer specializing in Math and Science Tutoring

Fanwood, NJ

5.0
(30 ratings)

Rudy N.

From Algebra to Calculus and Physics, at your service!

Scotch Plains, NJ

4.8
(12 ratings)

Samuel H.

Professional Math Tutor

Brooklyn, NY

4.9
(103 ratings)