How do I determine which number is bigger as $n$ gets sufficiently large, $2^n$ or $n^ {1000}$?  It seems to me it is a limit problem so I tried to tackle it that way.  $$\lim_{n\to \infty} \frac{2^n}{n^{1000}}$$  My thoughts are that, after some $n$, the numerator terms will be more than the terms in the denominator so we'll have something like $$\frac{\overbrace{2\times 2\times\cdots \times 2}^{1000\text{ factors}}}{n\times n \times \cdots \times n} \times 2^{n-1000}$$ At this point, I was thinking of using the fact that $2^n$ grows faster slower than $n!$ as $n$ gets larger so the limit, in this case, will be greater than $1$, meaning $2^n$ is bigger than $n^{1000}$ for sufficiently large $n$. This conclusion is really just a surmise based on a non-concrete formulation. Therefore, I'd appreciate any input on how to tackle this problem.
In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem.
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ bash dynamic (variable) variable names How can I add an empty directory to a Git repository? RedirectMatch and Query String Compare these two RedirectMatch's. The first doesn't work: RedirectMatch 302 ^/redirect\.php[?]page=(.+)$ http://somewhereelse.com/$1  Versus this, which will redirect to http://somewhereelse.com/?page=wherever: RedirectMatch 302 ^/redirect\.php([?]page=.+)?$ http://somewhereelse.com/$1  Does RedirectMatch only match the URI and not the query string? Apache's documentation is a bit vague in this area. What I'm trying to do is extract the page query parameter and redirect to another site using it. Is that possible with RedirectMatch or do I have to use RewriteCond + RewriteRule? Infusing olive oil with herbs I know that Dirichlet function has uncountable many discontinuities. 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How to remove Ubuntu from dual boot system with Windows? I wish to remove Ubuntu and I'm not quite sure how. I know it has something to do with partitions. I have 3 partitions. 1. OS [683 GB]  2. Recovery [14 GB] 3. Ubuntu 12.04 LTS (E:)  I know it says Ubuntu but when I installed Ubuntu I think I put 14 gigabytes for it so I think its the recovery one but I'm not sure. I could have sworn I put more than 700 megabytes for it.  I have googled this and I can't seem to find an answer. Any help would be nice. Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$ Where is the denoiser? Most games build on other games' successes and many are downright clones. Where is the limit of imitation before legal issues come into play? Is it down to literary or graphic work like characters and storyline that cause legal problems, or can someone actually claim to own gameplay mechanics?  There are so many similar clone games out there that the rules are probably very slack or nonexistent, but I'd like to hear the views of more experienced developers / designers.