Monday, August 24, 2015

calculus - Proving that $n over 2^n$ converges to $0$


I'm completely clueless on this one. I can easily calculate the limit using L'Hopital's rule, but proving that the series is converging to 0 is far more tricky.


$$a_n= {n \over 2^n}$$


Any help?


Answer



Using the fact that $2^n>n^2$ for $n > 4$, we have: $$0 \leq a_n < \frac{n}{n^2}=\frac{1}{n}.$$



Hence, $\displaystyle \lim_{n \to \infty}a_n=0.$


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