Question

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

Answer 1

`lim_(n->oo) root(n)n = 1`

Explanation:

As the terms of the sequence:

`a_n = root(n)n`

are positive, we can consider the sequence:

`b_n = ln a_n = ln root(n)n`

Using the properties of logarithms we have:

`b_n = ln root(n) n = (lnn)/n`

Now consider the function:

`f(x) = lnx/x`

if `lim_(x->oo) f(x)` exists, it must be the same as `lim_(n->oo) b_n`, since `b_(n) = f(n)`.

The limit `lim_(x->oo) lnx/x` is in the indeterminate form `oo/oo` and we can determine it using l'Hospital's rule:

`lim_(x->oo) ln/x = lim_(x->oo) (d/dx lnx) / (d/dx x) = lim_(x->oo) 1/x = 0`

and we have established that:

`lim_(n->oo) ln a_n = 0`

As `lnx` is a continuous function we can also state that:

`lim_(n->oo) ln a_n = ln (lim_(n->oo) a_n) = 0`

which implies:

`lim_(n->oo) a_n = 1`