I am currently trying to solve a sum that I need for my Probability class. The sum goes as follows: $$ \sum_{k=0}^\infty \binom{a+k-1}{k}r^k $$ where $a\in\mathbb{N}$ and $r\in [0,1]$. I have tried finding similar question on the site, but I could not find any.
I have tried calculating the common ratio, $r\frac{a+k}{k+1}$, but that does not seem to help me. I have looked through my Discrete mathematics $1$ notes in hopes that I can find a similar sum I have evaluated last semester, but to no avail. I'm not sure this even converges, it feels like it doesn't - even though this is for Probability class (there could be a chance I messed something up while getting to this sum).
Any help would be greatly appreciated. If necessary, I can post the entire problem I am trying to solve. Moreover, this is one of my first posts on the site, so if there are any tips/suggestions about formatting the question, those would also be welcome
Thank you in advance.
Note that $$ (1-r)^{-a}=\sum_{k=0}^\infty\binom{-a}{k}(-r)^k=\sum_{k=0}^\infty\binom{a+k-1}{k}r^k $$ by the negative binomial theorem (for $|r|<1$) since $$ \binom{-a}{k}(-1)^k=(-1)^k\frac{-a(-a-1)\dotsb(-a-k+1)}{k!}=\frac{a(a+1)\dotsb(a+k-1)}{k!}=\binom{a+k-1}{k} $$