Re: Discovered series result
Ok I figured out another way to solve this .
The expression is equivalent to
{sum n=k to infinity}C(n,k)x^(n-k)*(1-x)^(k+1)=1
Suppose you have a bias coin with probability x of landing heads and probability (1-x) of landing tails . You keep on flipping this coin infinitely many times until you've flipped k+1 tails . Certainly the probability of this happening is 1 which is the rhs of the equation . Or equivalently ,
you may hit k+1 tails on your (k+1)st turn ,
k+1 tails on your (k+2)nd turn
k+1 tails on your (k+3)rd turn
This is exactly the lhs of the equation .
Why didn't I think of this before ?
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