blocks|key|3578364|text|不！最终比基数固定的指数(2%5En和e%5En)增长更快，但n%5En比n！因为基数随着n的增加而增长。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|3578365|entityMap^0|0^^$0|@$1|2|3|4|5|6|7|D|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|E|8|@]|9|@]|A|$]]]|C|$]]

n! eventually grows faster than an exponential with a constant base (2^n and e^n), but n^n grows faster than n! since the base grows as n increases.

blocks|key|1980603|text|n!+=+n+*+(n-1)+*+(n-2)+*+...|type|unstyled|depth|inlineStyleRanges|offset|length|style|CODE|entityRanges|data|1980604|n%5En+=+n+*+n+*+n+*+...|1980605|在n%5En中，第一项之后的每一项都更大，所以n%5En增长得更快。|1980606|entityMap^0|0|S|0|0|L|0|1|3|0^^$0|@$1|2|3|4|5|6|7|L|8|@$9|M|A|N|B|C]]|D|@]|E|$]]|$1|F|3|G|5|6|7|O|8|@$9|P|A|Q|B|C]]|D|@]|E|$]]|$1|H|3|I|5|6|7|R|8|@$9|S|A|T|B|C]]|D|@]|E|$]]|$1|J|3|-4|5|6|7|U|8|@]|D|@]|E|$]]]|K|$]]

<code>n! = n * (n-1) * (n-2) * ...</code>

<code>n^n = n * n * n * ...</code>

Every term after the first one in <code>n^n</code> is larger, so n^n will grow faster.

blocks|key|1852138|text|n%5En比n!更大--有关更好的解释，请参阅@AlexQueue的答案。|type|unstyled|depth|inlineStyleRanges|offset|length|style|CODE|entityRanges|data|1852139|对于其他情况，请继续阅读：|1852140|阶乘函数确实比指数函数渐近增长，但从什么时候开始还不清楚。例如，对于n=5和k=10，阶乘5!=120仍然小于10%5E5=10000。为了找出阶乘函数何时开始变大，我们必须进行一些快速的数学分析。|1852141|我们使用Stirling公式和基本的对数运算：|1852142|log_k(n!)+~+n*log_k(n)+-+n*log_k(e)

k%5En+=+n!
log_k(k%5En)+=+log_k(n!)
n*log_k(k)+=+log_k(n!)
n+=+log_k(n!)

n+~+n*log_k(n)+-+n*log_k(e)
1+~+log_k(n)+-+log_k(e)
log_k(n)+-+log_k(e)+-+1+~+0
log_k(n)+-+log_k(e)+-+log_k(k)+~+0
log_k(n/(e*k))+~+0

n/(e*k)+~+1
n+~+e*k|code-block|syntax|javascript|1852143|因此，一旦n的大小几乎达到k的3倍，阶乘函数将开始大于指数函数。对于大多数现实场景，我们将使用较大的n值和较小的k值，因此在实践中，我们可以假设阶乘函数严格大于指数函数。|1852144|entityMap^0|0|3|4|2|0|0|Y|3|12|4|19|6|1J|A|0|0|0|5|1|D|1|1E|1|1K|1|0^^$0|@$1|2|3|4|5|6|7|U|8|@$9|V|A|W|B|C]|$9|X|A|Y|B|C]]|D|@]|E|$]]|$1|F|3|G|5|6|7|Z|8|@]|D|@]|E|$]]|$1|H|3|I|5|6|7|10|8|@$9|11|A|12|B|C]|$9|13|A|14|B|C]|$9|15|A|16|B|C]|$9|17|A|18|B|C]]|D|@]|E|$]]|$1|J|3|K|5|6|7|19|8|@]|D|@]|E|$]]|$1|L|3|M|5|N|7|1A|8|@]|D|@]|E|$O|P]]|$1|Q|3|R|5|6|7|1B|8|@$9|1C|A|1D|B|C]|$9|1E|A|1F|B|C]|$9|1G|A|1H|B|C]|$9|1I|A|1J|B|C]]|D|@]|E|$]]|$1|S|3|-4|5|6|7|1K|8|@]|D|@]|E|$]]]|T|$]]

<code>n^n</code> grows larger than <code>n!</code> -- for an excellent explanation, see the answer by @AlexQueue.

For the other cases, read on:

Factorial functions do asymptotically grow larger than exponential functions, but it isn't immediately clear when the difference begins. For example, for <code>n=5</code> and <code>k=10</code>, the factorial <code>5!=120</code> is still smaller than <code>10^5=10000</code>. To find when factorial functions begin to grow larger, we have to do some quick mathematical analysis.

We use Stirling's formula and basic logarithm manipulation:

<pre><code>log_k(n!) ~ n*log_k(n) - n*log_k(e)

k^n = n!
log_k(k^n) = log_k(n!)
n*log_k(k) = log_k(n!)
n = log_k(n!)

n ~ n*log_k(n) - n*log_k(e)
1 ~ log_k(n) - log_k(e)
log_k(n) - log_k(e) - 1 ~ 0
log_k(n) - log_k(e) - log_k(k) ~ 0
log_k(n/(e*k)) ~ 0

n/(e*k) ~ 1
n ~ e*k
</code></pre>

Thus, once <code>n</code> reaches almost 3 times the size of <code>k</code>, factorial functions will begin to grow larger than exponential functions. For most real-world scenarios, we will be using large values of <code>n</code> and small values of <code>k</code>, so in practice, we can assume that factorial functions are strictly larger than exponential functions.

blocks|key|1980663|text|我想向你展示一个更图形化的方法来非常容易地证明这一点。我们将使用除法来绘制一个函数，它将很容易地向我们展示这一点。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1980664|让我们使用一个基本而枯燥的除法函数来解释除法的一个性质。|1980665|​|1980666|📷|atomic|offset|length|1980667|1980668|随着a的增加，对该表达式的评估也会增加。随着b的减少，该表达式的求值也会减少。|1980669|使用这个想法，我们可以根据我们预期的增加和减少绘制一张图，并比较哪个增加得更快。|1980670|在我们的例子中，我们想知道指数函数是否会比阶乘函数增长得更快，或者反之亦然。我们有两种情况，一个常量对应一个可变指数，一个变量对应一个变量阶乘，一个变量对应一个可变指数，一个变量对应一个变量阶乘。|1980671|用Desmos绘制这些工具的图表(没有从属关系，它只是一个很好的工具)，向我们展示了：|1980672|常数到可变指数的图，vs变量阶乘|1980673|1980674|1980675|1980676|虽然最初看起来指数表达式增加得更快，但它达到了一个点，它不再增加得那么快，相反，阶乘表达式增加得更快。|1980677|变量对变量指数的图，vs变量阶乘|1980678|1980679|1980680|1980681|虽然它最初看起来更慢，但它开始快速上升超过这一点，因此我们可以得出结论，指数增长必须比阶乘更快。|1980682|entityMap|0|IMAGE|mutability|IMMUTABLE|imageUrl|https://ask.qcloudimg.com/http-save/yehe-900000/0edf52d38c84400e9f7d46093470fbc9.png|imageAlt|1|https://ask.qcloudimg.com/http-save/yehe-900000/865f5f870f0619e53ea94c335c02f9bd.png|2|https://ask.qcloudimg.com/http-save/yehe-900000/c5b664ccdb4bf12d64686407414caa66.png^0|0|0|0|0|1|0|0|0|0|0|0|0|0|0|0|1|1|0|0|0|0|0|0|1|2|0|0|0^^$0|@$1|2|3|4|5|6|7|1K|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|1L|8|@]|9|@]|A|$]]|$1|D|3|E|5|6|7|1M|8|@]|9|@]|A|$]]|$1|F|3|G|5|H|7|1N|8|@]|9|@$I|1O|J|1P|1|1Q]]|A|$]]|$1|K|3|E|5|6|7|1R|8|@]|9|@]|A|$]]|$1|L|3|M|5|6|7|1S|8|@]|9|@]|A|$]]|$1|N|3|O|5|6|7|1T|8|@]|9|@]|A|$]]|$1|P|3|Q|5|6|7|1U|8|@]|9|@]|A|$]]|$1|R|3|S|5|6|7|1V|8|@]|9|@]|A|$]]|$1|T|3|U|5|6|7|1W|8|@]|9|@]|A|$]]|$1|V|3|E|5|6|7|1X|8|@]|9|@]|A|$]]|$1|W|3|G|5|H|7|1Y|8|@]|9|@$I|1Z|J|20|1|21]]|A|$]]|$1|X|3|E|5|6|7|22|8|@]|9|@]|A|$]]|$1|Y|3|Z|5|6|7|23|8|@]|9|@]|A|$]]|$1|10|3|11|5|6|7|24|8|@]|9|@]|A|$]]|$1|12|3|E|5|6|7|25|8|@]|9|@]|A|$]]|$1|13|3|G|5|H|7|26|8|@]|9|@$I|27|J|28|1|29]]|A|$]]|$1|14|3|E|5|6|7|2A|8|@]|9|@]|A|$]]|$1|15|3|16|5|6|7|2B|8|@]|9|@]|A|$]]|$1|17|3|-4|5|6|7|2C|8|@]|9|@]|A|$]]]|18|$19|$5|1A|1B|1C|A|$1D|1E|1F|-4]]|1G|$5|1A|1B|1C|A|$1D|1H|1F|-4]]|1I|$5|1A|1B|1C|A|$1D|1J|1F|-4]]]]

I want to show you a more graphical method to very easily prove this. We're going to use division to graph a function, and it will show us this very easily.

Let's use a basic and boring division function to explain a property of division.

<a href="https://i.stack.imgur.com/LipA5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LipA5.png" alt="division with variables a and b"></a>

As a increases, the evaluation of that expression also increases. As b decreases, the evaluation of that expression also decreases.

Using this idea, we can plot a graph based on what we expect to increase, and expect to decrease, and make a comparision as to which increases faster.

In our case, we want to know whether exponential functions will grow faster than factorials, or vice versa. We have two cases, a constant to a variable exponent vs. a variable factorial, and a variable to a variable exponent vs a variable factorial.

Graphing these tools with Desmos (no affiliation, it's just a nice tool), shows us this:

<h3>Graph of a constant to variable exponent, vs variable factorial</h3>

<a href="https://i.stack.imgur.com/vM0hx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vM0hx.png" alt="graph 1"></a>

Although it initially seems that the exponential expression increases faster, it hits a point where it no longer increases as fast, and instead, the factorial expression is increasing faster.

<h3>Graph of a variable to variable exponent, vs variable factorial</h3>

<a href="https://i.stack.imgur.com/uEbeB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uEbeB.png" alt="graph 2"></a>

Although it initially seems to be slower, it begins to rise rapidly past that point, therefore we can conclude that the exponential must be increasing faster than the factorial.

Which function grows faster, exponential (like 2^n, n^n, e^n etc) or factorial (n!)? 
Ps: I just read somewhere, n! grows faster than 2^n.

Which function grows faster, exponential or factorial?

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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哪个函数增长更快，指数(如2^n，n^n，e^n等)还是阶乘(n!)？附言:我只是在某处读到了，n！增长速度超过2^n。

问指数函数和阶乘函数哪个增长更快？
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问指数函数和阶乘函数哪个增长更快？
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