Approximation of sum of the first binomial coefficients for fixed N I'd like to compute $\sum_{i=0}^k {{N}\choose{i}}$. Is there a computable approximation for that?
 A: 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.
A: A well-known upper bound, for $k\le N/2$, is
$$ \sum_{i=0}^k {N\choose i} \le 2^{N H(k/N)},$$
where $H$ is the binary entropy function
$$ H(x) = -x\log_2(x)-(1-x)\log_2(1-x).$$
This bound was sharpened in Lemma 5 of
https://www.sciencedirect.com/science/article/pii/S0012365X12000544 (
Gottlieb, Lee-Ad; Kontorovich, Aryeh; Mossel, Elchanan, VC bounds on the cardinality of nearly orthogonal function classes, Discrete Math. 312, No. 10, 1766-1775 (2012). ZBL1242.05050.)
