Sunday, December 23, 2018

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


Please show how to prove that lim


Answer



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


Then use L'Hopital's rule: lim_{x\to\infty} x/2^x has indeterminate form \infty/\infty. Taking the limit of the quotient of derivatives we get lim_{x\to\infty} 1/(ln2\cdot 2^x)=0. Thus lim_{x\to\infty} x/2^x=0 and so n/2^n\to 0 as n\to\infty.



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