Binomial Coefficients proof?
Hello I need some help proving this fact. I've tried proving it in different ways but I'm stuck at the moment. I know the even terms of the binomial coefficients minus the odd terms=0 but the m factor is messing me up here.
(m choose 1)-2(m choose 2)+3(m choose 3)+...+m(m choose m)((-1)^(m+1))=0
for m>=2.
Thanks you in advance
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Start with the binomial theorem (1 + x)^m = Σ(k = 0 to m) C(m, k) x^k.
Differentiate both sides:
m(1 + x)^(m-1) = Σ(k = 1 to m) k C(m, k) x^(k-1).
Let x = -1:
m(1 + (-1))^(m-1) = Σ(k = 1 to m) k C(m, k) (-1)^(k-1)
==> 0 = Σ(k = 1 to m) k C(m, k) (-1)^(k+1), since (-1)^(k+1) = (-1)^2 * (-1)^(k-1).
I hope this helps!