Let $\alpha \in (0, 1)$. What is the asymptotic growth of the following function? $$\sum_{x = 0}^{\alpha n} {n \choose x}$$
I am aware that ${n \choose \alpha n}$ grows asymptotically as a constant times $\left( \frac{1-\alpha}{\alpha} \right)^{\alpha n} \left( \frac{1}{1 - \alpha} \right)^n$, but I can't seem to find the asymptotic growth of the previous function I had mentioned. I tried taking an integral from $x = 0$ to $\alpha n$ of the quantity $$\left( \frac{n-x}{x} \right)^x \left( \frac{n}{n-x} \right)^n$$ but I can't find the closed form expression of this integral. Does anybody have a better idea?