Let $n,k$ be positive integers. From the binomial theorem, we know that


For what $i$ (in terms of $n,k$) is it true that the sum of the first $i$ terms is approximately $\frac1k$? In other words, for what $i$ is it that


I'm not sure how to start with approximating this binomial sum.


The problem can be stated as finding a certain quantile for a binomial distribution.

If $n$ is large enough, such a distribution can be approximated by a normal distribution and the answer be formulated in terms of $\Phi^{-1}$, i.e. the inverse function of the CDF of a normal distribution.


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