# On the Limit of Stirling's Approximation

I have recently proven the following curious identity: For real $x \geqslant 1$, \begin{align} \lfloor x \rfloor! = x^{\lfloor x \rfloor} e^{1-x} e^{\int_{1}^{x} \text{frac}(t)/t \ dt} \end{align} where $\text{frac}(x)$ denotes the fractional part (function) of $x$. A proof follows from an application of Legendre's Theorem on the order of a prime divisor of the factorial function and an identity involving the de Bruijn function.

An asymptotic expansion of the integral as $x \to \infty$ yields Stirling's Approximation, hence the title of the post. Is this exact result known in the literature?

Thanks!

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+1: Very nice! Is the proof too involved to post here? –  Aryabhata Feb 18 '12 at 18:54
The proof of a much more general identity is a bit involved and too long to post here. The identity above is a special case. –  user02138 Feb 18 '12 at 22:08
@Aryabhata: See my recent post below. –  user02138 Feb 29 '12 at 21:19
Thank you for notifying me. I already did (and voted) :-) –  Aryabhata Feb 29 '12 at 21:21

That works. There are two parts. Write $x = n + \delta,$ with $n \in \mathbb Z$ and $0 \leq \delta < 1.$ Your identity $$\lfloor x \rfloor! = x^{\lfloor x \rfloor} e^{1-x} e^{\int_{1}^{x} \text{frac}(t)/t \ dt}.$$ becomes $$n! = (n + \delta)^n \; e^{1-n} \; e^{-\delta} \; e^{\int_{1}^{n} \text{frac}(t)/t \ dt} \; e^{\int_{n}^{n + \delta} \text{frac}(t)/t \ dt}$$ As the left hand side $n!$ is independent of $\delta,$ we need the factor that depends on $\delta$ to be a constant (but depending on $n$). Note that $${\int_{n}^{n + \delta} \text{frac}(t)/t \ dt} = \delta + n \log \left( \frac{n}{n+\delta} \right)$$ So the factor $$(n + \delta)^n \; e^{-\delta} \; e^{\int_{n}^{n + \delta} \text{frac}(t)/t \ dt} = (n + \delta)^n \; e^{-\delta} \; e^{\delta} \; \left( \frac{n}{n+\delta} \right)^n = n^n$$ is indeed independent of $\delta,$ and your identity is $$n! = n^n \; e^{1-n} \; e^{\int_{1}^{n} \text{frac}(t)/t \ dt} .$$ This is rather cute. Proof by induction: it is true for $n=1,$ both sides are just $1.$ In the induction step, we get the right hand side with $(n+1)$ as $$RHS_{n+1} = (n+1)^{n+1} \; e^{-n} \; e^{\int_{1}^{n} \text{frac}(t)/t \ dt} \; e^{\int_{n}^{n+1} \text{frac}(t)/t \ dt}.$$ The final factor is what we got with $\delta = 1,$ that is $${\int_{n}^{n+1} \text{frac}(t)/t \ dt} = 1 + n \log \left( \frac{n}{n+1} \right)$$ So $$RHS_{n+1} = (n+1) \; (n+1)^n \; e^{-n} \; e^{\int_{1}^{n} \text{frac}(t)/t \ dt} \; \cdot e \cdot \; \left( \frac{n}{n+1} \right)^n$$ and $$RHS_{n+1} = (n+1) \; n^n \; e^{1-n} \; e^{\int_{1}^{n} \text{frac}(t)/t \ dt}$$ and the induction hypothesis says $$RHS_{n+1} = (n+1) \; n!$$

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Hi Will. Thanks for your observation. Once one has the identity, it can be proved by induction, as you've just demonstrated. My question is whether the identity has appeared in the literature. –  user02138 Feb 18 '12 at 22:05
@user02138, I really don't now. –  Will Jagy Feb 19 '12 at 0:02
+1. I'm not terribly interested in whether or not this is a previously published result, I just wanted to see the proof, and this one is so nice! –  Matt Calhoun Feb 19 '12 at 13:40
@MattCalhoun, thank you. –  Will Jagy Feb 19 '12 at 20:31

Although my question above concerns whether or not the posted identity is a known result, I'll add the following elementary and direct proof (without recourse to Legendre's Theorem or the de Bruijn function needed for a more general version).

Proposition For real $x \geqslant 1$, the following identity holds \begin{align} \lfloor x \rfloor! = x^{\lfloor x \rfloor} e^{1-x} e^{\int_1^{x} \frac{\text{frac}(t)}{t} \ dt}. \end{align}

Proof. For each positive integer $n$, the function $f(t) = \tfrac{\text{frac}(t)}{t}$ equals $1 - \tfrac{n}{t}$ on $[n,n+1]$. Thus, \begin{align} \int_{1}^{n} f(t) \ d t & = \sum_{k = 1}^{n-1} \int_{k}^{k+1} (1 - \tfrac{k}{t}) \ dt \\ & = n - 1 + \log n! - n \log n. \end{align} By splitting the following integral into two pieces, we compute \begin{align} \int_{1}^{x} f(t) \ d t & =\int_{1}^{\lfloor x \rfloor} f(t) \ d t + \int_{\lfloor x \rfloor }^{x} f(t) \ dt \\ & = \lfloor x \rfloor - 1 + \log \lfloor x \rfloor! - \lfloor x \rfloor \log \lfloor x \rfloor + \int_{\lfloor x \rfloor}^{x } f(t) \ d t. \end{align} We need only to compute the last integral, \begin{align} \int_{\lfloor x \rfloor}^{x } f(t) \ dt & = \int_{0}^{\text{frac}(x) } \frac{\text{frac}(t + \lfloor x \rfloor)}{t + \lfloor x \rfloor} \ d t \\ & = \int_{0}^{ \text{frac}(x) } \frac{t}{t + \lfloor x \rfloor } \ d t \\ & =\text{frac}(x) + \lfloor x \rfloor \log \lfloor x \rfloor - \lfloor x \rfloor \log x \end{align} by a change of variables from $t$ to $t + \lfloor x \rfloor$. Hence, together these identities yield \begin{align} \int_{1}^{x} f(t) \ dt & = x - 1 + \log \lfloor x \rfloor! - \lfloor x \rfloor \log x, \end{align} which completes the proof of the claim.

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+1 very nice!!! –  draks ... Feb 29 '12 at 21:18
@draks: Thanks. ;) –  user02138 Feb 29 '12 at 21:19