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The precise asymptotics of the inequality partition function $n! \ge \frac{n^n}{e^n}$, but p(n)$are, as far as I don't knowof a hands-on proof, totally inaccessible by finitary tools. More generally the methods of complex analysis are a great way to obtain incredibly precise asymptotics on many finitary sequences of concrete interest, i.e. trees satisfying certain properties or sequences related to the running times of algorithms, etc. A related example is comparing the elementary and non-elementary proofs of the prime number theorem; my recollection is that the elementary proof is quite a bit more involved. 1 Using Cauchy's integral formula one very easily obtains the inequality$n! \ge \frac{n^n}{e^n}\$, but I don't know of a hands-on proof. More generally the methods of complex analysis are a great way to obtain incredibly precise asymptotics on many finitary sequences of concrete interest, i.e. trees satisfying certain properties or sequences related to the running times of algorithms, etc.