$$ \prod\limits_{i=0}^{n}{1 \over x + i} = {1 \over n!}\sum\limits_{i=0}^{n}{(-1)^i {n \choose i} \over x + i} $$
For example when \(n = 2\),
$$ {1 \over x(x + 1)(x + 2)} = {1 \over 2}({1 \over x} - {2 \over x + 1} + {1 \over x + 2}) $$
I have no proof yet.
Now I have the idea that it takes to prove it. Note that for example 1/x(x+1)(x+2) - 1/(x+1)(x+2)(x+3) = (x+3-x)/x(x+1)(x+2)(x+3) = 3/x(x+1)(x+2)(x+3). Generalize the formula and apply it inductively to get the result.
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