TITLE: Approximation of sum of the first binomial coefficients for fixed N
QUESTION [10 upvotes]: I'd like to compute $\sum_{i=0}^k {{N}\choose{i}}$. Is there a computable approximation for that?

REPLY [16 votes]: One of the more convenient and popular approximations of the sum is
$$\frac{2^{nH(\frac{k}{n})}}{\sqrt{8k(1-\frac{k}{n})}} \leq \sum_{i=0}^k\binom{n}{i} \leq 2^{nH(\frac{k}{n})}$$
for $0< k < \frac{n}{2}$, where $H$ is the binary entropy function. (The upper bound is exactly what Aryeh Kontorovich mentions.) You can find its proof in many textbooks, but probably I first learned it from Chapter 10 of The Theory of Error-Correcting Codes by MacWilliams and Sloane.
Also, this post on MO asks a similar question and is a good resource for the sum in my opinion. You can find several other useful bounds there.