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

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Answer 1 · 2017-02-08T17:18:48

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

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#