# 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?

## 2 Answers

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.

• Nice, I didn't know about the upper bound. That's probably as well as you can do, with such simple estimates. – Aryeh Kontorovich Feb 5 '17 at 15:41
• I like the word "popular". :) – Wolfgang Feb 6 '17 at 15:03

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 http://www.sciencedirect.com/science/article/pii/S0012365X12000544

• The sharpening adds a prefactor of $0.98$; the authors comment that a smaller prefactor is possible with a more careful/detailed proof. The paper is also available, open access, on arXiv: arxiv.org/abs/1007.4915; there it is Lemma 7.1, in Section 7 – Sam T Aug 6 '19 at 9:59