MHB Why Does This Summation Simplify to a Power of p?

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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 series converges to the desired power of $p$. This transformation highlights the relationship between the binomial coefficients and the probability term. Understanding this simplification is crucial for grasping the underlying principles of combinatorial identities in probability theory. The discussion emphasizes the significance of the binomial series in simplifying complex summations.
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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?
 
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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?
Hint: Use the binomial series $$(1+x)^\alpha = \sum_{k=0}^\infty {\alpha\choose k}x^k,$$ with $x = -(1-p)$ and $\alpha = 1-r.$
 
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