Re: Discovered series result
Oh, yes. Nice observation: negative binomial.
{sum n=k to infinity}C(n,k)x^(n-k)*(1 - x)^(k+1)
= {sum n=0 to infinity}C(n+k,k)x^n*(1 - x)^(k+1)
= {sum n=0 to infinity}((n + k)!/(n!k!))x^n*(1 - x)^(k+1)
= {sum n=0 to infinity}((n + k)(n + k - 1)...(k + 1)/n!)x^n*(1 - x)^(k+1)
= (1 - x)^(k+1){sum n=0 to infinity}((-k - 1)(-k - 2)...(-k - n))/n!)(-x)^n
= (1 - x)^(k+1){sum n=0 to infinity}C(-k-1,n)(-x)^n
= (1 - x)^{k+1}(1 + (-x))^{-k-1}
= 1
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