Finding the sum of series involving factorials and combinatorics in denominator and/or numerator

Question

I'm currently finishing my Calculus 2 course in university, our professor gave us the last test to do it at home, like an open test, and I'm stuck with a couple of problems involving some series, I'd appreciate if you could help me out here on how to proceed and reach a possible solution,

1. $\sum_{n=0}^\infty \frac{1}{\binom{r+n+1}{r+1}}, r\in \mathbb{R}$

2. $\sum_{n=0}^\infty \frac{1}{(n+a)(n+a+1)\cdot\cdot\cdot(n+a+r)}$, $a+n\neq0\space\forall n, r\in \mathbb{R}$

Both problems ask me to find the sum of the series, I'm really clueless on how to procced here, I've tried to decompose the numerator so it becomes a telescopic series, but no luck in that. Any insight here would be appreciate it, Thanks.

Answer 1

HInt

For the second question note the decomposition $$ \frac{1}{(n+a)\dotsb(n+a+r)}=\frac{1}{r}\left(\frac{1}{(n+a)\dotsb(n+a+r-1)}-\frac{1}{(n+a+1)\dotsb(n+a+r)}\right) $$