Asymptotics of $ f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k} $

Question

Define $$ f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k} $$ for some fixed constant $c$ (say, $0<c<1/2$). What are the asymptotics of $f_c(n)$ as $n\to\infty$?

It seems that this should be simple but I can't seem to figure it out. I searched the standard references (MathWorld, Wikipedia, DLMF, etc.) but didn't see any identities for this type of sum.

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Edit: It seems, numerically, that with $$ g(c)=\lim_{n\to\infty}\log(f_c(n))/n $$ that $g(c)$ exists and is positive for $c>0$. (Of course $g(1)=\log2.$) Is this so, and can $g(c)$ be computed efficiently? Or even more courageously, what is the error in approximating $f_c(n)$ as $\exp(n\cdot g(c))$?

Answer 1

I presume you mean $\lfloor c n\rfloor$ rather than $\lfloor c k \rfloor$ in that sum.

$$f_c(n)/2^n = \mathbb P(S_n \le \lfloor c n \rfloor)$$

where $S_n$ is a binomial random variable with parameters $n$ and $1/2$.
By Large Deviations theory,

$$\lim_{n \to \infty} \dfrac1n \log f_c(n) = - c \log c - (1-c) \log(1-c) $$

Answer 2

There is no identity, as $n$ is growing to infinite, $cn$ is to. So the limit is the infinte sum of the coeficient which are a growing infinite sequence, the limit is infinite.