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Really stumped on this one, anyone good at divergence of series?

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hey there is a of getting a bigger picture it is a little a hard to read for me at least just asking..
\[\sum_{3}^{∞} \frac{ 1 }{ n \ln n \sqrt{(\ln n)^2-1} }\]

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Other answers:

It is kinda small, but it looks like you'd want to use comparison test. I think you might be able to use integral test also since you are workingn with log, and you know the values must be positive.
any suggestions for what to compare it with? I redid the equation for readability
one sec
try this website
Use integral test.
@godorovg this is for series, not functions in vector fields.
sorry it has been a while for me. I am really sorry man..
You can use to evaluate the series just as you would an improper integral. Try it to see what you get. you are going to be using u-substitution.
I am trying to find the right way to help here
is this what I need to look at? I am making sure I do this right is all..
If you are lookng at that site: Example one would be a similar example to the problem you are working on now.
okay let me review this
The thing that bothers me, is for the integral test to apply, the derivative must be less than 0, which it doesn't appear to be
that is what I was think also it must be + not - for this test to work as well
according to what I have read on the website at least.
yeah, back to square one I guess
no it does work here because when you look the rule of squares and taking LN that is squared would be at 1 at least not zero.
also given that when infinty goes to 3 zero would not be included right?
so I am thinking the series it is itself would 1,2,3 and so on leaving out and the end result than would be postive so therefore test works
Am I making sense to you?
because the limit would be 3 in this case
you still with me??
so let me make sure I do this right, the integral is lim n->inf sec^-1(lnx) evaluated from 3 to t right?
if you come up with zero at any time test fails..
can you check the answer after you are finished?
i got pi/2 -arcsec(ln3), implying that the sum converges
that looks right
I realise that this might be moot, but how about checking the convergence of \[\frac{1}{n(\ln n)^2}\] instead? Then it can be shown to be convergent, and by the limit comparison test, you'll be able to show that your original series was also convergent.
did you check the answer or can even do that depending if this an odd or even in the book you are working out of? So, how are you doing so far??

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