Discovered series result
Sometimes a probability problem affords one solution by some general mathematical technique, and another solution purely from probability considerations, and this provides insight into a general mathematical problem. I just stumbled upon such a problem which enables me to sum the following series:
sum{n = k to infinity} C(n,k)*x^n = (x^k)/(1-x)^(k+1)
for 0 < x < 1.
At first glance this may appear to be a more familiar series, until you realize that k is held constant, and the sum is over n. Do not confuse this with the more familiar binomial expansion in which n is constant and the sum is over k.
Can you prove this? Is this series result well-known? Will I be famous? [img]/images/graemlins/smile.gif[/img]
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