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Stirling's series is an asymptotic series for $n!$ given as follows:

$$n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 +\frac{1}{12n}+\frac{1}{288n^2} - \frac{139}{51840n^3} -\frac{571}{2488320n^4}+ \cdots \right).$$

As an asymptotic series, the large $n$ error is asymptotically equal to the first omitted term. For example, the error using just the first term is $O(1/n)$.

This is a purely qualitative statement, and doesn't tell us much about how big the error actually is. The constant hiding in the $O(1/n)$ could, a priori, be very large. It happens the the approximation is quite good, but we know that only because we've computed $n!$ exactly and compared it to the asymptotic expansion.

Is there any way to prove a quantitative guarantee on the accuracy of this series? I know only how to derive the explicit leading order correction $1/12n$ and non-explicitly prove the existence of the rest of the series, but the method I'm familiar with, from Simon's A Comprehensive Course in Analysis, volume 2B, is not explicit enough to track the error even to this first term. The other coefficients are derived here without any discussion of the error.

Since we're in a simple and specific situation it seems like we should be able to be very precise.

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    Look up the Euler Maclaurin summation formula (applied to $\sum_{i=1}^n f(i)$ with $f(x) = \log x$) with its remainder term. Just as Taylor polynomial approximations to a function have an integral and differential form of the remainder, the Euler Maclaurin summation formula has integral and differential form of its remainder. I've seen the integral form more often. It involves an integral of a higher derivative of the function being summed times periodized Bernoulli polynomial (divided by a factorial). Even with just a few terms the error is easy to bound above and is often small. – 2017-01-30
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    @KCd Thanks, I'm not familiar with that method. Is it possible to get the entire asymptotic expansion that way. The notes I'm finding with Google only do the leading order. – 2017-01-30
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    Yes, that's what the Euler Maclaurin summation formula is all about. (Historically Euler and Maclaurin didn't have an error term in their version of the summation formula -- the idea of error bounds even for power series was too advanced for that time -- and just added up the terms until they would "start to diverge".) Read derivations of the Euler Maclaurin summation formula in books and how it applies to Stirling's formula for $\log(n!)$. For example, this is discussed in Hairer/Wanner's "Analysis by its History". – 2017-01-31
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    @KCd Thanks! I'll definitely check up on that. – 2017-01-31
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    It is also discussed at the end of Graham/Knuth/Patashnik's "Concrete Mathematics" (both the Euler Maclaurin summation formula with error term and applying it to Stirling's series for $\log n!$). For the time being this book is available online at http://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf. – 2017-01-31
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    @KCd Ah, I see. The asymptotic series in both of those references is slightly different than the one I'm asking about: $\log n !$ versus $n !$, and I don't think it's possible to convert error bounds from one to the other. – 2017-01-31
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    The relation you asked about is asymptotic in the multiplicative sense. If in the asymptotic series for $\log n!$ you can show for some $n$ that $S(n) - \varepsilon \leq \log n! \leq S(n) + \varepsilon$, where $S(n)$ is the Stirling series, then $e^{-\varepsilon} \leq n!/e^{S(n)} \leq e^{\varepsilon}$, so you have a multiplicative approximation (ratio near 1 if $\varepsilon$ is small). That's the most you could expect from an asymptotic relation. – 2017-01-31
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    @kcd Yes. But the series estimated in your link is not the series I give in my question. Am I missing something? – 2017-01-31
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    One is just the exponential of the other. See the asymptotic relations for $n!$ and $\ln n!$ at https://en.wikipedia.org/wiki/Stirling's_approximation. Check $\exp(x/12 - x^3/360) = 1 - x/12 +x^2/288 - 139x^3/51840+\cdots$ and set $x = 1/n$. – 2017-01-31
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    There is a detailed discussion of the error term in the following paper: https://arxiv.org/pdf/1310.0166.pdf (see also http://dx.doi.org/10.1017/S0308210513001558) – 2018-03-20

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