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Is there a closed form expression for the following sum:

$$\sum_{k=0}^s (-1)^k {n \choose k}, $$

where $s \in \{0,1,2,...,n\}$

-- which is basically the first $s$ terms in the alternating binomial coefficients series. I know if $s=n$, the sum is zero, but I cannot figure out a general closed form expression in terms of $s$.

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Yes there is: $$(-1)^s {n-1 \choose s}$$

Proof by induction, for example, is straightforward.

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  • $\begingroup$ Thanks, the formula works - but how do you get the formula? $\endgroup$ – Argonne Dec 1 '15 at 1:36
  • $\begingroup$ @Argonne Are you asking (a) how do you prove it given what it is, or (b) how would you go about finding out what it is if you didn't know? $\endgroup$ – Sharkos Dec 1 '15 at 11:36

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