Integral Test for Limits clarification

It is my understanding that for the integral test to apply, the series a must be decreasing, positive, and continuous over [n, ∞) but one of the problems I've encountered seems to break the first rule, yet my textbook still says it converges. I don't know what I am doing incorrectly.


Summation of [ln(n)]/(n²) starting at n=1, counting to infinity
f(x) = [ln(n)]/(n²)
f'(x) = [1 - 2ln(n)]/(n³)

0 = 1 - 2ln(n)
n = e^.5 = 1.645


When you do a sign test on either side, you discover that after, to the right, f'(x) is negative, which to the left it is positive. This would leave me to believe that you could not apply the integral test here, yet the book I'm using put this in with the section about the integral test and says it diverges, which leads me to believe I'm missing something.

Thanks

PS sorry for the lack of formatting, I tried to get it to work, but it wasn't cooperating.

it doesn't have to be decreasing from the beginning I mean from n=1 all the way to infinity ,,, but it is sufficient to do so for n sufficiently large ...

in your example it is increasing ,, then decreasing , but as you stated it has to be decreasing from (n,infinity )

Can you clarify?

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Originally Posted by hayjude99

Can you clarify?

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if you have the function

on the interval

Can you tell me whether the integral test conditions can be applied to this function?

Yes, they can. Are you saying that basically the first few terms don't matter, so as long as it eventually decreases, the integral test applies?

Also in the OP, I meant to say that the book says it converges. xP sorry. Though I suppose the question is still valid of whether or not I can use the integral test.

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Originally Posted by hayjude99

Are you saying that basically the first few terms don't matter, so as long as it eventually decreases, the integral test applies?

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absolutely correct !

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Originally Posted by hayjude99

It is my understanding that for the integral test to apply, the series a must be decreasing, positive, and continuous over [n, ∞)

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See that the condition [n, ∞) states that , so we have to find n such that all the conditions apply .

Best regards ...

Quote:

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Originally Posted by hayjude99

PS sorry for the lack of formatting, I tried to get it to work, but it wasn't cooperating.

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You can quote my reply to see how it works (Happy)

Oh alright, thanks :)
I shall!