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Determine whether the sequence converges or diverges. If it converges, find the limit....

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user6978788 | Student, Undergraduate | Honors

Posted March 27, 2013 at 5:53 PM via web

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Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
`lim_(n->oo)(ln(6n))/(ln(12n))=`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 27, 2013 at 6:12 PM (Answer #1)

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You need to evaluate the given limit, such that:

`lim_(n->oo) (ln(6n))/(ln(12n)) = oo/oo`

The indetermination oo/oo requests for you to use l'Hospital's theorem, such that:

`lim_(n->oo) (ln(6n))/(ln(12n)) = lim_(n->oo) ((ln(6n))')/((ln(12n))')`

`lim_(n->oo) ((ln(6n))')/((ln(12n))') = lim_(n->oo) (6/(6n))/(12/(12n))`

Reducing duplicate factors yields:

`lim_(n->oo) (ln(6n))/(ln(12n)) = lim_(n->oo) 1 = 1`

Hence, since the sequence `(ln(6n))/(ln(12n))` converges to the limit 1, hence, the sequence is convergent.

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