Re: partial sum of binomial coefficients
From: Richard Mathar (mathar_at_amer.strw.leidenuniv.nl)
Date: 12/09/04
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Date: Thu, 9 Dec 2004 19:52:00 +0000 (UTC)
twittoy@bgumail.bgu.ac.il (Yochai) writes:
>...
>I would like to know if the is a close formula for a partial sum of
>binomial coefficient.
>
>For example, let n be some large enough positive integer. And let eps
>be some positive rational number smaller that 1 (e.g. 1/3 or 1/7).
>
>Is there exist a formula (or even a good estimation) on the sum:
>C(n,0) + C(n,1) + C(n,2) + ... + c(n, eps*n) ??
>..
If one inserts p=1/2 into the sum formula of 26.5.24 in the Abramowitz/Stegun
book http://jove.prohosting.com/skripty/ for the binomial distribution, one
sees that the complement of this sum (ie the sum from eps*n to n)
is an incomplete beta function. Further up in chapter 26.5 one then finds
series expansions and asymptotic expansions for these. The full sum
is of course known by eq 3.1.6, 2^n.
Richard J. Mathar
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