# How do you use the integral test to determine if #Sigma lnn/n^2# from #[1,oo)# is convergent or divergent?

The series:

If we use:

Calculating the first derivative:

so all the hypotheses of the integral test theorem are satisfied and the series is convergent if the integral:

also converges.

Let's calculate the indefinite integral by parts:

So:

That is the integral is convergent and so is the series.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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