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

Answer 1

The series

#sum_(n=0)^oo ln(ln (n))#

is not convergent.

A necessary condition for any series:

#sum_(n=0)^oo a_n #

to converge is that:

#lim_(n->oo) a_n = 0#
In fact if we consider the #n#-th partial sum:
#s_(n-1) = sum_(k=0)^(n-1) a_k#

we have:

#(1) s_n = s_(n-1) +a_n#

Now if the series is convergent this means that:

#lim_(n->oo) s_n = L# with #L in RR#

and clearly this implies that also:

#lim_(n->oo) s_(n-1) = L#
but as from #(1)#:
#lim_(n->oo) s_n = lim_(n->oo) s_(n-1) + lim_(n->oo) a_n#

we have:

#L = L + lim_(n->oo) a_n#

which implies:

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

Now as:

#lim_(n->oo) ln(ln (n)) = oo#

the series is not convergent.

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Answer 2

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.

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

  2. 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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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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