Today we prove a remarkable asymptotic result regarding the binomial coefficients. Namely, let . We will show
Before getting to the proof, I should say that this is most definitely not my creation; I believe the first publication of this proof may be credited to Polya (but I am not sure). In any case, I was shown this proof at MSU in my Putnam training seminar. It was presented by Dr. Ignacio Uriarte-Tuero. Any unintended errors introduced are mine alone.
To start, we use the definition of to write
Now for , we want to examine the power of which appears in the fraction. So, for instance, appears times in the numerator and twice in the denominator from , appears times in the numerator and four times in the denominator from and , etc. We get the formula
We now multiply by in a clever way. Noting that we obtain
as the denominator has total exponent . We can now write
We want to examine . Taking logarithms, we get
We now recognize this expression as a Riemann sum of the function on the interval . (Ok, it is not exactly a Riemann sum; we should be dividing by , not . But in the limit this will not matter.) Taking limits, it follows that
Evaluating the integral is now easy via integration by parts; it turns out that
and thus and .
And we’re done! As always, thanks for reading!