negative binomial

For any integer m>=2, prove mathematically that the following infinite series sum converges to the limit 1

Sum_(n=m...inf) C(n-1, m-1)/2^n =1

我只会用归纳法证明，不知各位有何良方？


consider the negative binomial.

suppose you flip a fair coin. let X be the time you observe m successes.
then
P(X=n)=C(n-1,m-1)*(1/2)^n for all n>=m.

this form a distribution. of course the sum of the probability mass
functions is 1.



Use the Taylor expansion of 1/(1-x)^m at x=0, write the coefficient of x^k
as C(m-1, m+k-1), set x = 1/2, you will get this identity immediately.
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