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

Answer 1

The series:

#sum_(n=3)^oo (-1)^n/ln(lnn)#

is convergent.

We have that:

#lim_(n->oo) 1/ ln(ln(n)) = 0#

Consider the function:

#f(x) = 1/ln(ln(x))#

As:

#f'(x) = -1/(ln(lnx))^2 1/lnx 1/x < 0# for #x > 1#
the function is strictly decreasing in #[1,+oo)# and thus:
#1/ln(ln(n+1)) < 1/ln(ln(n))#

thus the series:

#sum_(n=3)^oo (-1)^n/ln(lnn)#

is convergent based on Leibniz' theorem.

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Answer 2

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:

  1. The terms of the series are decreasing.
  2. 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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Answer from HIX Tutor

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.

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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