I would like to find some approximation or upper & lower bounds on the next simple expression: \begin{align} \sum_{i = 0}^{k} \binom{h}{i} \qquad h \geq k \end{align} But I need this approximation/bounds to be computable in polylogarithmic time - i.e. in: \begin{align} O(\log_{}^{c} h) \qquad c \in N \end{align}
If you still follow you may be intrested in my specific problem: given two numbers: \begin{align} n, k \in N \qquad n \geq 2^k \end{align} find in polylogarithmic time (as function of n) a number: \begin{align} h \in N \end{align} such that: \begin{align} \sum_{i = 0}^{k} \binom{h-1}{i} < n \le \sum_{i = 0}^{k} \binom{h}{i} \end{align} It is obvious that: \begin{align} \lceil \log_{2}n \rceil \leq h \leq n-1 \end{align} Thanks in advance.