from Stirling, we know the asymptotic for single binomial coefficients as $k,n$ go to infinity. Now I'm trying to estimate the order of a sum of binomial coefficients where $k$ ranges in an interval between two constant fractions of $n$. In particular, if $0\le a\le b\le 1$, $$ \sum_{k=\lfloor an\rfloor}^{\lfloor bn\rfloor} {n\choose k} = ? $$ and in particular, how does it compare with $2^n$?
2026-04-06 07:36:38.1775460998
asymptotic for sums of binomials
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