Sums of binomial coefficients - accuracy of CLT approximation?

Question

I was trying to figure out the asymptotics of the sum $$S(n) = \sum_{k=0}^{\lfloor \alpha n \rfloor} \binom{n}{k}$$ for $0 < \alpha < 1/2$ (actually, the specific case of interest for me is $\alpha = 1 - \log_3(2)$). Can I use Central Limit Theorem and get something along the lines of $$S(n) \approx \frac{1}{2} \left( 1 + \operatorname{erf} \left( (\alpha - 0.5)\sqrt{2n} \right) \right) * 2^n $$ for $n$ sufficiently large? How bad is the approximation for large $n$, and what might these asymptotics look like in terms of "elementary" functions?

Answer 1

$\binom{n}{\lfloor\alpha n\rfloor}$ (and therefore $S(n)$, as we'll see) doesn't have a "smooth" asymptotics. While $$\frac{\Gamma(1+n)}{\Gamma(1+\alpha n)\Gamma\big(1+(1-\alpha)n\big)}\underset{n\to\infty}{\asymp}\frac{\big(\alpha^\alpha(1-\alpha)^{1-\alpha}\big)^{-n}}{\sqrt{2\pi\alpha(1-\alpha)n}},$$ an "error" of $\mathcal{O}(1/n)$ in $\alpha$ is not negligible.

But if we retain such an expression in the asymptotics, we're fine: $$\color{blue}{\lim_{n\to\infty}\binom{n}{\lfloor\alpha n\rfloor}^{-1}S(n)=\frac{1-\alpha}{1-2\alpha}.}$$

An easy way to see this is to use the formula (I'm taking it from this answer): $$\sum_{k=0}^{m}\binom{n}{k}a^{n-k}(1-a)^k=(n-m)\binom{n}{m}\int_0^a x^{n-m-1}(1-x)^m~dx\qquad(0\leqslant m<n)$$

If we put $a=1/2$, $m=\lfloor\alpha n\rfloor$, and substitute $x=(1-y)/2$, we obtain $$\binom{n}{\lfloor\alpha n\rfloor}^{-1}S(n)=(n-\lfloor\alpha n\rfloor)\int_0^1(1-y)^{n-\lfloor\alpha n\rfloor-1}(1+y)^{\lfloor\alpha n\rfloor}\,dy;$$ after $y=t/n$, we see that the RHS tends to $(1-\alpha)\int_0^\infty e^{-(1-\alpha)t}e^{\alpha t}\,dt$ (as $n\to\infty$).