On Ramanujan's approximation, $n!\sim \sqrt{\pi}\big(\frac ne\big)^n\sqrt [6]{(2n)^3+(2n)^2+n+\frac 1{30}}$

Question

Over here I discovered that Ramanujan gave the following factorial approximation, better than Stirling's formula:

$$n!\sim \sqrt{\pi}\left(\frac ne\right)^n\sqrt [6]{(2n)^3+(2n)^2+n+\frac 1{30}}$$

such that the error term decreases rapidly as $n\to \infty$. In other words, $$\lim_{n\to\infty}\cfrac{n!}{\sqrt{\pi}\left(\frac ne\right)^n\sqrt [6]{8n^3+4n^2+n+\frac 1{30}}}=1$$ Just to add, Stirling's formula is: $$n!\sim \sqrt{2\pi n}\left(\frac ne\right)^n$$ so somehow Ramanujan was able to turn $2n$ into $\sqrt [3]{8n^3+4n^2+n+\frac 1{30}}$. Notice that $2n=\sqrt [3]{8n^3}$ so the important expression is $4n^2+n+\frac 1{30}$.

Does anybody know how he got this result? Or is this another one of his mysterious results...

Answer 1

Edited after @Gary's comment.

This is question I could have been asking for years. I wonder if the basis of this magnificent approximation has been published anywhere.

What I have seen (the problem is that I do not remember where) are approximations built as $$n!\sim \sqrt{\pi}\left(\frac ne\right)^n\sqrt [2k]{P_k(n)}$$ where remains the mystery of the $\sqrt [2k]{.}$. The first case I saw is Gosper approximation.

The coefficients of the polynomials were obtained taking the logarithms of both sides and identified with the Stirling series. So, what was obtained are $$P_1(n)=2n+\frac 13$$ $$P_2(n)=4n^2 + \frac{{4}}{3}n + \frac{2}{9}$$ $$\color{red}{P_3(n)=8 n^3+4 n^2+n+\frac{1}{30}}$$ and for sure, with the tools we have today, we could continue for ever $$P_4(n)=16 n^4+\frac{32 }{3}n^3+\frac{32 }{9}n^2+\frac{176 }{405}n-\frac{128}{1215}$$ $$P_5(n)=32 n^5+\frac{80 }{3}n^4+\frac{100 }{9}n^3+\frac{178}{81}n^2-\frac{95}{972}n+\frac{2143}{40824}$$ $$P_6(n)=64 n^6+64 n^5+32 n^4+\frac{128 }{15}n^3+\frac{8 }{15}n^2+\frac{8 }{105}n+\frac{596}{1575}$$

for more and more accuracy.

For example, for $n=5$, the magic formula given by the great Ramanujan gives $120.000147066$ while the last given here leads to $120.000000406$.

Edit

Staying with the power $\frac 16$ from Ramanujan, we could extend it as $$P_3(n)=8 n^3+4 n^2+n+\frac{1}{30}\left(1+\sum_{i=1}^\infty \frac {a_i}{n^i}\right)$$ and the sequence of the first $a_i$'s is $$\left\{-\frac{11}{8},\frac{79}{112},\frac{3539}{6720},-\frac{9511}{13440},-\frac{30 153}{71680},\frac{233934691}{212889600},\frac{3595113569}{5960908800},\cdots\right\}$$

Answer 2

The asymptotic expansion of the factorial can be expressed as:

$$ n! \sim \left( \frac{n}{\mathrm{e}} \right)^n \sqrt{2\pi n} \sum_{k=0}^\infty (-1)^k \frac{\gamma_k}{n^k}, $$

where $\gamma_k$ represents the Stirling coefficients (refer to here for more information). The first three coefficients are $\gamma_0=1$, $\gamma_1=-\frac{1}{12}$, and $\gamma_2=\frac{1}{288}$. Now, if a function possesses an asymptotic expansion, then its sixth power also has one, given by raising the original expansion to the power of $6$. Hence,

$$ \left( \frac{n!}{\left( \frac{n}{\mathrm{e}} \right)^n \sqrt{2\pi n}} \right)^6 \sim \left( \sum_{k=0}^\infty (-1)^k \frac{\gamma_k}{n^k} \right)^6 \sim 1 + \frac{1}{2n} + \frac{1}{8n^2} + \frac{1}{240n^3} - \frac{11}{1920n^4} + \ldots, $$

which implies

\begin{align*} n! & \sim \left( \frac{n}{\mathrm{e}} \right)^n \sqrt{2\pi n} \sqrt[6]{1 + \frac{1}{2n} + \frac{1}{8n^2} + \frac{1}{240n^3} - \frac{11}{1920n^4} + \ldots} \\ & = \left( \frac{n}{\mathrm{e}} \right)^n \sqrt{\pi} \sqrt[6]{8n^3 + 4n^2 + n + \frac{1}{30} - \frac{11}{240n} + \ldots}\;. \end{align*}

Hence, Ramanujan's approximation is essentially a manipulation of the standard asymptotic expansion of the factorial. It is notably more accurate than the leading-order asymptotics (known as Stirling's formula) as it incorporates additional terms from the asymptotic expansion. It is worth mentioning that although the standard asymptotic expansion (and Ramanujan's as well) is divergent, for large values of $n$, the initial terms decrease in magnitude. A minimum occurs around $\left\lfloor {2\pi n} \right\rfloor$. Truncating the series at this point results in exponentially accurate approximations (for more details, refer again to the aforementioned paper).