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The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function \begin{equation} \sum_{k=0}^mC_n^kr^k, \quad m<n \end{equation} for fixed $n$ and $r$, and both $m$ and $n$ are integers. Are there any notable properties for this function? Any literature references?

In particular, do any good closed-form approximations exist for this partial sum of the binomial theorem?

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You are asking for tail estimates for the binomial distribution. An introduction can be found on Wikipedia - this has pointers to further work.

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Here is one interesting property:

MR2147385
Ostrovskii, Iossif V. On a problem of A. Eremenko. Comput. Methods Funct. Theory 4 (2004), no. 2, 275–282.

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