Closed form sum for the series given below? Does the following series have a closed form sum?
$$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$

Pascal's Identity states that $\dbinom{r+i}{r} + \dbinom{r+i}{r+1} = \dbinom{r+i+1}{r+1}$. 
Hence, $\displaystyle\sum_{i = 1}^{n}\dbinom{r+i}{r} = \sum_{i = 1}^{n}\left[\dbinom{r+i+1}{r+1} - \dbinom{r+i}{r+1}\right]$. 
This sum telescopes to $\dbinom{r+n+1}{r+1} - \dbinom{r+1}{r+1} = \dbinom{r+n+1}{r+1} - 1$.
Now, add $\dbinom{r}{r} = 1$ to get  $\displaystyle\sum_{i = 0}^{n}\dbinom{r+i}{r} =\dbinom{r+n+1}{r+1}$. 
Note: This is commonly referred to as the Hockey-Stick Identity. 
