Is there any way to derive Stirlings formula that only requires some undergraduate knowledge of calculus, real analysis and perhaps some identitets involving the gamma function, maybe Wallis product, and things along those lines? If not, and I know this is a rather vague question, what is the simplest but still sufficiently rigorous way of deriving it? I had a look at Stirling's formula: proof? but the comments seems quite messy. Wikipedia was not particularly helpful either since I have not learned about Laplace's method, Bernoulli numbers or the trapezoidal rule.
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