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By using The Ratio Test determine if the series is convergent: `sum` `a^n/n^3`

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Rocky52 | Student, Undergraduate | (Level 2) Honors

Posted April 29, 2013 at 5:02 AM via web

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By using The Ratio Test determine if the series is convergent:

`sum` `a^n/n^3`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 29, 2013 at 5:37 AM (Answer #1)

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Using the Ratio Test gives the convergence of the series if `lim_(n->oo)((a^(n+1))/((n+1)^3))/((a^n)/(n^3)) < 1`

`lim_(n->oo)a^(n+1 - n)/(((n+1)^3)/n^3)`

= `lim_(n->oo)a/((n/n+1/n)^3)`

= `lim_(n->oo)a/((1+1/n)^3)`

when `n->oo, 1/n->0`

= a

As the value of a is not known, nothing can be said about a < 1.

It is not possible to determine if the given series is convergent using the Ratio Test.

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