How do you determine if #a_n=(1+n)^(1/n)# converge and find the limits when they exist?

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Answer 1 · 2017-11-08T15:00:30

The series converge to #=1#

Let #y=(1+n)^(1/n)#

Taking logarithms

#lny=ln((1+n)^(1/n))=1/nln(1+n)#

Therefore,

#y=e^(1/nln(1+n))#

So,

#lim_(n->oo)u_n=lim_(n->oo)e^(1/nln(1+n))#

Apply the limit chain rule

#lim_(n->oo)(1/nln(1+n))=lim_(n->oo)(1/(1+n))# ( by L'Hôpital's rule)

#=0#

Therefore,

#lim_(n->oo)e^(1/nln(1+n))=lim_(n->oo)e^0=1#