A relevant fact, which has not been mentioned, not explicitly at least: the integral remainder formula in the Taylor expansion, $$\sum_{i=0}^k{N\choose i}=2^N- N{N-1\choose k}\int_0^1(1+t)^{N-k-1}(1-t)^kdt=$$ $$ =2^N\Big[1- N{N-1\choose k}\int_0^{\frac12}\Big(\frac12+s\Big)^{N-k-1}\Big(\frac12-s\Big)^kds\Big].$$

A relevant fact, which has not been mentioned, not explicitly at least: the integral remainder formula in the Taylor expansion, $$\sum_{i=0}^k{N\choose i}=2^N- N{N-1\choose k}\int_0^1(1+t)^{N-k-1}(1-t)^kdt=$$ $$ =2^N\bigg[1- N{N-1\choose k}\int_0^{\frac12}\Big(\frac12+s\Big)^{N-k-1}\Big(\frac12-s\Big)^kds\bigg].$$