How do you determine whether the sequence #a_n=n/(ln(n)^2# converges, if so how do you find the limit?

Answer 1

The series:

#sum_(n=2)^oo n/(lnn)^2#

is divergent.

Note: The question is not mentioning it expressly, but in the answer, I assume the series starts from #n=2# as #a_n# is undefined for #n=1#

A necessary condition for any series to converge is Cauchy's condition that:

#lim_(n->oo) a_n=0#

To verify whether this condition is met, first we note that

#a_n = f(n)#
where #f(x) = x/ln(x)^2# is a continuous function for # x in (1,+oo)#
So if the limit #lim_(x->oo) f(x)# exixst it must be the same as #lim_(n->oo) a_n#

Now:

#lim_(x->oo) f(x) = lim_(x->oo) x/(lnx)^2#
is the inderminate form #oo/oo# and we can solve it using the l'Hospital's rule (twice):
#lim_(x->oo) x/(lnx)^2 = lim_(x->oo) 1/((2lnx)/x) = lim_(x->oo) x/(2lnx) = lim_(x->oo) 1/(2/x) = lim_(x->oo) x/2 = +oo#

Thus also:

#lim_(n->oo) n/(lnn)^2 = +oo#

which implies that the series is not convergent.

As it has positive terms, the series is then divergent.

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Answer 2

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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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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