TITLE: Convergence of the probability of RV with negative binomial distribution
QUESTION [2 upvotes]: I am seeking an answer to this question.

Given the number of tools successfully produced $R$ before the $yth$ failure follows a negative binomial distribution $(y,p)$. 
$$
P(R=r)=\begin{pmatrix}r+y-1\\
r
\end{pmatrix}(1-p)^{y}p^{r}
$$
Prove that $P(R=r)$ converges to $0$ as $y$ increases.

I can see that I need to prove that the factorial ratio 
$
\begin{pmatrix}r+y-1\\
r\end{pmatrix}
$ is going to infinity slower than $(1-p)^y$ is going to $0$, as $y$ increases.

REPLY [2 votes]: Let $f(y)=\binom{r+y-1}{r}(1-p)^yp^r$, where for this answer $r$ is fixed. 
Look at $\frac{f(y+1)}{f(y)}$. This is equal to 
$$\frac{(r+y)!}{y!r!}\cdot \frac{(y-1)!r!}{(r+y-1)!}(1-p),$$
which simplifies to 
$$\frac{r+y}{y}(1-p).$$
For fixed $r$, the fraction  $\frac{r+y}{y}$ has limit $1$ as $y\to\infty$. If $p\ne 0$, then by the Ratio Test the series 
$$\sum_1^\infty f(y)$$
converges, and therefore $\lim_{y\to\infty} f(y)=0$.
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