Using Limit Comparison Test to Determine Convergence/Divergence

[izen]

Homework Statement

Using the limit comparison test to determine convergence or divergence


(1) \sum^{\infty}_{n=1} \frac{(ln(n))^{3}}{n^{3}}


(2) \sum^{\infty}_{n=1} \frac{1}{\sqrt{n} ln(x)}


(3)\sum^{\infty}_{n=1} \frac{(ln(n))^{2}}{n^{\frac{3}{2}}}

Homework Equations

The Attempt at a Solution

I have problem find thing the another series (Bn) compare with An. First I choose Bn when 'n' grows large to represent An

The first question I chose Bn = \frac{1}{n^{3}}
it is the p-series and p>1 so it converges

and then
lim_{n\rightarrow\infty} \frac{\frac{ln(n)^{3}}{n^{3}}}{\frac{1}{n^{3}}}

lim_{n\rightarrow\infty} ln(n)^{3} = \infty

which is wrong so I change to choose Bn = \frac{1}{n^{2}}

lim_{n\rightarrow\infty} \frac{\frac{ln(n)^{3}}{n^{3}}}{\frac{1}{n^{2}}}

lim_{n\rightarrow\infty} \frac{ln(n)^{3}}{n}= 0


So I can conclude that An is convergent by limit comparison

but why I cannot use Bn = \frac{1}{n^{3}} ?

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(2) and (3) are the same as my first question

(2) \sum^{\infty}_{n=1} \frac{1}{\sqrt{n} ln(x)}

I chose Bn = \frac{1}{n^{1/2}} but after I took the limit of An/Bn. I got 0. In the answer shows that I have to choose Bn = \frac{1}{n^{}}

(3)\sum^{\infty}_{n=1} \frac{(ln(n))^{2}}{n^{\frac{3}{2}}}

I chose Bn = \frac{1}{n^{3/2}} after I took the limit of An/Bn. I got \infty. In the answer shows that I have to choose Bn = \frac{1}{n^{5/4}}

Please give some comments and advices

Thank you

 

[mfb]

but why I cannot use Bn = \frac{1}{n^{3}} ?

The idea is to compare your series with a bigger one with known convergence. 1/n^3 is not bigger, therefore it does not work.

The same happens in the other direction, if you want to show divergence.

 

[izen]

Thank you for the comments

The idea is to compare your series with a bigger one with known convergence. 1/n^3 is not bigger, therefore it does not work.

The same happens in the other direction, if you want to show divergence.

\sum^{\infty}_{n=1} \frac{(ln(n))^{3}}{n^{3}}

The series (Bn) that we choose must represent the original series (An) and we recognize converges or diverges from Bn. (We don't know from the original). the problem is...

How do I know that \frac{1}{n^3}is too small \frac{(ln(n))^{3}}{n^{3}} and\frac{1}{n^2}not too big for \frac{(ln(n))^{3}}{n^{3}} and why not choose\frac{1}{n}. still not get it.

 

[mfb]

1/n does not give a converging series.
You could choose 1/n^3/2 or 1/n^5/2 or something similar, if you like, but 1/n² is easier to study.