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I am trying to bound values of very large factorials (N>500) without using Sterling's formula.

For large N, I found that n!>$100^n$ is a relatively good lower bound for the factorial. After trying for a while, I can't find a higher bound for factorials that is actually relatively close to the true value of the factorial.

Any suggestions for tighter lower and higher bounds are appreciated. Thanks

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    $\begingroup$ $100^n$ is just a lower bound for sufficiently large $n$. So ist $1000^n$ and in general $N^n$. Such functions are not growing fast enough. You must look at $n^n$ or better $\left(\frac ne\right)^n$ but this comes very close to Striling's formula. $\endgroup$ – M. Winter Feb 23 '17 at 18:34
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    $\begingroup$ What is the reason for avoiding Stirling's approximation? $\endgroup$ – Qudit Feb 23 '17 at 18:35
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    $\begingroup$ For $n\ge 269$, you have $100^n<n!<n^n$. Not very tight, but easy provable (induction is already suffcient) $\endgroup$ – Peter Feb 23 '17 at 18:47
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    $\begingroup$ @Sumant If you want an approximation that is asymptotically correct, you can't avoid Stirling. But there are better elementary bounds than the one you give above. $\endgroup$ – Qudit Feb 23 '17 at 18:48
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    $\begingroup$ If you are happy with the approximation of $100^n$ for lets say $n=500$, then you can choose $n!\approx n^n$ for every $n$. This will be a relative good approximation in the same sense that $100^{500} \approx 500!$ is relatively good. $\endgroup$ – Peter Feb 23 '17 at 18:54

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