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Convexity

Let S n =[ n;...

Question

Let $S_{n} = (_{0}^{n}) - (_{1}^{n - 1}) + (_{2}^{n - 2}) - \dots$
Note that the series does terminate. Recall that $^{p} C_{q} = 0$ if $p < q$
$S_{2009}$ equals

-1

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Solution

$S_{n} = S_{n + 1} \forall n$
$S_{1} = 1, S_{2} = 0, S_{3} = - 1, S_{4} = - 1, S_{5} = 0, S_{6} = 1$

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