The summation $\sum_{x=0}^{\infty} \binom{x+r-2}{r-2}(1-p)^{x}$ simplifies to $p^{1-r}$ through the application of the binomial series. By substituting $x = -(1-p)$ and $\alpha = 1-r$ into the binomial series formula, the relationship is established. This simplification is crucial for understanding the convergence properties of the series in relation to the parameter $p$.
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Hint: Use the binomial series $$(1+x)^\alpha = \sum_{k=0}^\infty {\alpha\choose k}x^k,$$ with $x = -(1-p)$ and $\alpha = 1-r.$pp123123 said:I came across some summation but have no idea how to simplify it.
$\sum_{x=0}^{\infty} \binom{x+r-2}{r-2}(1-p)^{x}=p^{1-r}$
Why is it so?