closed form expression for the sum of the first s items of alternating binomial coefficients

Question

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 $s$.

Does this answer your question? Truncated alternating binomial sum — ViHdzP, Dec 19 '19 at 07:29

score 4 · Answer 1 · answered Dec 01 '15 at 01:26

4

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

Proof by induction, for example, is straightforward.

answered Dec 01 '15 at 01:26

not all wrong

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Thanks, the formula works - but how do you get the formula? – Argonne Dec 01 '15 at 01:36
@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? – not all wrong Dec 01 '15 at 11:36

closed form expression for the sum of the first s items of alternating binomial coefficients

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