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By the inequality here or using integration I can get the following lower bound for $\log(n!)$. How can I get a better lower bound for $\log(n!)$?


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The log's in your question are superfluous. You are essentially lower-bounding $n!$. There are very good lower-bounds, notably Sterling's approximation gives $n! > \sqrt{2\pi n}(n/e)^n$. – Fixee Mar 27 '11 at 6:41
I think you can develop that bound by looking at the Euler-McLaurin summation of $\sum_{k=1}^n \log k$. – Raskolnikov Mar 27 '11 at 9:10
Re Fixee's comment, see…. – Did Mar 27 '11 at 9:45

It might be worth spelling out the geometric picture behind Stirling's approximation.

On the one hand, take the integral $\int_1^{n} \log t dt$ and bound it from below by $n-1$ trapezoids, where the vertices of the $k$-th trapezoid are $(k,0)$, $(k+1, 0)$, $(k, \log k)$ and $(k+1, \log (k+1))$.

So $$\int_1^n \log t dt \geq \sum_{k=1}^{n-1} \frac{\log k + \log (k+1)}{2} = \log n! - \frac{\log n}{2}$$ That gives $$(n \log n -n) - (1 \log 1 - 1) + 1/2 \log n = (n+1/2) \log n - n + 1\geq \log n!.$$

To get a bound the other way, we'll approximate $\int_{3/2}^{n+1/2} \log t dt$. We again use a sequence of trapezoids. Our trapezoids have vertices at $(k-1/2, 0)$, $(k+1/2, 0)$, $(k-1/2, \log k - s)$ and $(k+1/2, \log k+s)$, for $2 \leq k \leq n$. Here $s$ is chosen so that the line through $(k \pm 1/2, \log k \pm s)$ is the tangent line to $\log t$ at the point $(k, \log k)$. (Explicitly, $s = 1/(2 \log k)$, but we don't need this.) This gives the bound $$\int_{3/2}^{n + 1/2} \log t dt \leq \sum_{k=2}^n \log k = \log n!.$$

So $$(n+1/2) \log (n+1/2) - (n+1/2) - (3/2) \log (3/2) + 3/2 \leq \log n!$$ or $$ \log n! \geq (n+1/2) \log (n+1/2) - n + 1-(3/2) \log (3/2) = $$ $$(n+1/2) \log n - n + (n+1/2) \log (1+1/(2n)) + 1- (3/2) \log (3/2) \geq$$ $$(n+1/2) \log n -n + (n+1/2) (1/(2n) - 1/(8n^2)) + 1- (3/2) \log (3/2) \geq $$ $$(n+1/2) \log n - n +3/2 - (3/2) \log (3/2) + 1/(8n).$$

Between the second and third line, I have bounded $\log (1+x)$ below by $x-x^2/2$, the first two terms of the alternating series for $\log$. Note that $1 > \log \sqrt{2 \pi} \approx 0.91894 > (3/2) - (3/2) \log (3/2) \approx 0.89180$, so our results are compatible with the greater precision given by Stirling's formula.

I should make pictures but I'm lazy, so I'll leave this community wiki and hope that someone else does.

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