# No closed form for the partial sum of ${n\choose k}$ for $k \le K$?

In Concrete Mathematics, the authors state that there is no closed form for $$\sum_{k\le K}{n\choose k}.$$ This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the book).

How do they know this is true?

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Actually, there is a "closed form", but it involves the Gaussian hypergeometric function. Thus, it boils down again to whose definition of "closed form" are we using... –  Ｊ. Ｍ. Aug 8 '12 at 16:26
I'm guessing the Concrete Mathematics authors had some specific result in mind, and my question is hoping to understand how one can show this expression does not have a closed form for whatever definition of "closed form" they were using. –  Tyler Aug 8 '12 at 16:32
Read the middle paragraph on page 228: "If we apply...is not summable in hypergeometric terms." –  Byron Schmuland Aug 8 '12 at 16:35