Lower and upper bounds for (sums of) binomial coefficients

Abstract

Apparently there is no closed form for the partial sum of a row of Pascal's triangle. In this paper lower and upper bounds for binomial coefficients and their sums are deduced. In the case of single coefficients these bounds differ only by a constant factor which is arbitrarily close to 1 for sufficiently large n. In the case of sums the gap between lower and upper bound is larger but still small enough to be useful in some applications. The upper bound obtained for sums is somewhat better than that resulting from a Chernoff bound.