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

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

1 Answer
Jul 3, 2018

1

Explanation:

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