# For sufficiently large $n$, Which number is bigger, $2^n$ or $n^{1000}$? [duplicate]

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.

• Exponential terms grow faster than polynomial terms for sufficiently large $n$. Also, $2^n$ does not grow faster than $n!$. – greelious Dec 6 '18 at 15:58
• I mixed up the two, thanks. I'll edit it. – E.Nole Dec 6 '18 at 16:00
• See related question/answers – farruhota Dec 6 '18 at 16:56

Note that$$\frac{2^{n+1}}{2^n}=2\text{ whereas }\frac{(n+1)^{1\,000}}{n^{1\,000}}=\left(1+\frac1n\right)^{1\,000}.$$Since$$\lim_{n\to\infty}\left(1+\frac1n\right)^{1\,000}=1<2,$$you have that$$\left(1+\frac1n\right)^{1\,000}<\frac32$$if $$n$$ is large enough.It's not hard to deduce from this that $$2^n>n^{1\,000}$$ if $$n$$ is large enough.
• I'd like to clarify what the first line says about the two expressions. What I can deduce is that the purpose of the first line is to show that $2^n$ is increasing at a constant rate whereas $n^{1000}$ starts to slow down in terms of increase rate. Is that so? – E.Nole Dec 6 '18 at 16:20
• It says that, after a certain point, the sequence $(n^{1\,000})_{n\in\mathbb N}$ grows slower than the sequence $(2^n)_{n\in\mathbb N}$. – José Carlos Santos Dec 6 '18 at 16:29