sequences and series - Proof that $lim_{ntoinfty} nleft(frac{1}{2}right)^n = 0$

Please show how to prove that $$\lim_{n\to\infty} n\left(\frac{1}{2}\right)^n = 0$$

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$_{x\to\infty} 1/($ln$2\cdot 2^x)=0$. Thus lim$_{x\to\infty} x/2^x=0$ and so $n/2^n\to 0$ as $n\to\infty$.