#lim_(n -> oo) n^(1/n)=?# for #n in NN# ?

Question

Given #x_n={""^nsqrt(n)}_(n=1)^oo#
thus #x_1=1,x_2=sqrt2,x_3=""^3sqrt(3),...#
prove: #lim_(n -> oo) x_n=1#

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Answer 1 · 2018-07-03T01:04:23

1

#f(n)=n^(1/n) implies log(f(n))=1/n log n#

Now

#lim_{n -> oo}log(f(n)) = lim_{n -> oo} log n/n#
#qquadqquadqquad = lim_{n -> oo} {d/(dn) log n}/{d/(dn) n} = lim_{n-> oo}(1/n)/1=0#

Since #log x# is a continuous function, we have

#log (lim_{n to oo}f(n))=lim_{n to oo} log(f(n)) = 0 implies#

#lim_{n to oo}f(n)=e^0=1#