Limits of sequences and series

Question

How do I go about proving limits for given sequences and series, for example how would i prove that the limit when x tends to infinity of n^(1/n)=1?

Answer 1

lim_n→∞(n^1/n) has the form ∞⁰.  This is an indeterminate form.

 

Let y = n^1/n

 

ln(y) = ln(n^1/n) = (1/n)ln(n) = (lnn)/n

 

    lim_x→∞(lnx/x) has the form ∞/∞

 

    By L'Hopital's Rule, lim_x→∞(lnx/x) = lim_x→∞[(1/x)/1) = 0

 

So, lim_n→∞(lny) = lim_n→∞(ln(n)/n)) = 0

 

Therefore, lim_n→∞(n^1/n) = lim_n→∞(y)

 

                                    = lim_n→∞(e^lny) = e⁰ = 1