Thursday, 1 January 2015

calculus - Suppose that liman=0 Prove that limnrightarrowinftyleft(1+anfracxnright)n=1.



Suppose that liman=0 Prove that limn(1+anxn)n=1




I was trying Leibniz theorem earlier but it was not working. Was I using the right one?


Answer



Using that (1+tn)net=exp(t)

as n with anx:=t you obtain that limn(1+anxn)n=limnexp(anx)=
Due to the continuity of the exponential function f(t)=et you can interchange the lim operator with the exp operator to conclude that =exp(limnanx)=exp(0)=e0=1


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