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Limit of sequence an an=radical n order(3^n+2010^n) n tend to `oo`

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xalal | Honors

Posted June 26, 2013 at 12:57 PM via web

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Limit of sequence an

an=radical n order(3^n+2010^n)

n tend to `oo`

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aruv | High School Teacher | Valedictorian

Posted June 26, 2013 at 1:15 PM (Answer #1)

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A general term of the sequence is `a_n=(3^n+2010^n)^(1/n)`

we want `lim_(n->oo)a_n=?`

we can write

`a_n={2010^n(1+(3/2010)^n}^(1/n)`

`=2010(1+(1/670)^n)^(1/n)`

`lim_{n->oo}a_n=2010`

because

`lim_{n->oo}(1+1/670^n)=1`

Thus answer is `lim_(n->oo)a_n=2010`

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