# How do you test the alternating series #Sigma (-1)^n/(ln(lnn))# from n is #[3,oo)# for convergence?

The series:

is convergent.

We have that:

Consider the function:

As:

thus the series:

is convergent based on Leibniz' theorem.

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To test the convergence of the alternating series Σ (-1)^n/(ln(lnn)) from n = 3 to infinity, you can use the Alternating Series Test.

First, check if the series satisfies the two conditions of the Alternating Series Test:

- The terms of the series are decreasing.
- The limit of the terms as n approaches infinity is 0.

To verify the first condition, observe that each term of the series (-1)^n/(ln(lnn)) alternates in sign and decreases as n increases.

To verify the second condition, find the limit of the absolute value of the terms as n approaches infinity. Use L'Hôpital's Rule if necessary.

If both conditions are satisfied, then the alternating series converges. If either condition fails, the test is inconclusive.

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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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