Sunday, December 23, 2018

sequences and series - Proof that limntoinftynleft(frac12right)n=0


Please show how to prove that limnn(12)n=0


Answer



Consider extending the sequence {n/2n} to the function f(x)=x/2x.


Then use L'Hopital's rule: limxx/2x has indeterminate form /. Taking the limit of the quotient of derivatives we get limx1/(ln22x)=0. Thus limxx/2x=0 and so n/2n0 as n.



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