How do you determine whether the sequence #a_n=n/(ln(n)^2# converges, if so how do you find the limit?
The series:
is divergent.
A necessary condition for any series to converge is Cauchy's condition that:
To verify whether this condition is met, first we note that
Now:
Thus also:
which implies that the series is not convergent.
As it has positive terms, the series is then divergent.
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To determine if the sequence ( a_n = \frac{n}{(\ln(n))^2} ) converges, we can use the limit comparison test. First, we find the limit of the sequence as ( n ) approaches infinity. If the limit exists and is finite, then the sequence converges. Otherwise, it diverges.
Let's find the limit:
[ \lim_{n \to \infty} \frac{n}{(\ln(n))^2} ]
We can use L'Hôpital's rule:
[ \lim_{n \to \infty} \frac{n}{(\ln(n))^2} = \lim_{n \to \infty} \frac{1}{2 \ln(n) \cdot \frac{1}{n}} ]
Applying L'Hôpital's rule again:
[ = \lim_{n \to \infty} \frac{1}{2 \cdot \frac{1}{n}} = \lim_{n \to \infty} \frac{n}{2} = \infty ]
Since the limit is infinity, the sequence ( a_n = \frac{n}{(\ln(n))^2} ) does not converge.
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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.
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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