How do you determine whether the sequence #a_n=ln(ln(n))# converges, if so how do you find the limit?
The series
is not convergent.
A necessary condition for any series:
to converge is that:
we have:
Now if the series is convergent this means that:
and clearly this implies that also:
we have:
which implies:
Now as:
the series is not convergent.
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To determine whether the sequence ( a_n = \ln(\ln(n)) ) converges, we can analyze its behavior as ( n ) approaches infinity. If the sequence approaches a finite limit as ( n ) goes to infinity, then it converges; otherwise, it diverges.
To find the limit of the sequence, if it converges, we evaluate the limit of ( \ln(\ln(n)) ) as ( n ) approaches infinity.

Consider ( \ln(\ln(n)) ) as ( n ) approaches infinity: [ \lim_{n \to \infty} \ln(\ln(n)) ]

Since ( \ln(n) ) grows unbounded as ( n ) approaches infinity, ( \ln(\ln(n)) ) approaches negative infinity as ( n ) goes to infinity.
Therefore, the sequence ( a_n = \ln(\ln(n)) ) diverges as ( n ) approaches infinity, and it does not have a finite limit.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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