Determining Convergence of Series with Logarithmic Terms

[ziggie125]

Homework Statement

Is the series convergent

1) \sum1/(n^2 * ln n)

and 2) which value of p does the series converge. \sum 1/(n*(ln n)^p)

The Attempt at a Solution

1)

I cannot see how the root method (\sqrt[n]{Cn}) would work, or the ratio test would work (cn+1)/cn

Unless you use the limit test and lim 1/(n^2 * ln n) = 1/infinity = 0 therefore the series converges. I didnt think that question would be that easy though.

2) Similiar to the first wouldn't it just be if P > 1.

 

[Mark44]

Homework Statement

Is the series convergent

1) \sum1/(n^2 * ln n)

and 2) which value of p does the series converge. \sum 1/(n*(ln n)^p)

The Attempt at a Solution

1)

I cannot see how the root method (\sqrt[n]{Cn}) would work, or the ratio test would work (cn+1)/cn

Unless you use the limit test and lim 1/(n^2 * ln n) = 1/infinity = 0

The test you are thinking of, I believe, is the nth term test for divergence. If lim a_n != 0, the series diverges. You cannot use this test to conclude a series converges.

A test that might be useful in this problem is the comparison test.

therefore the series converges. I didnt think that question would be that easy though.

2) Similiar to the first wouldn't it just be if P > 1.