What is limit in sequence an=n^(1/n)? n--->`oo`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the following limit of the given series `(a_n)` , such that:

`lim_(n->oo) n^(1/n) = oo^(1/oo) = oo^0`

You should re-write `n^(1/n)` such that:

`n^(1/n) = e^(ln (n^(1/n)))`

Using the power property of logarithms, yields:

`n^(1/n) = e^((1/n)(ln (n)))`

Taking the limit, yields:

`lim_(n->oo) n^(1/n) = lim_(n->oo) e^((1/n)(ln (n)))`

`lim_(n->oo) n^(1/n) = e^(lim_(n->oo)((ln n)/n))`

Since evaluating the limit `lim_(n->oo)((ln n)/n)` yields `lim_(n->oo)((ln n)/n) = 0` , you need to replace 0 for `(lim_(n->oo)((ln n)/n))` , such that:

`lim_(n->oo) n^(1/n) = e^0 = 1`

Hence, evaluating the given limit, under the given conditions,  yields `lim_(n->oo) n^(1/n) = 1.`

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