double summation binomial coefficient

Question

I have the following sum to evaluate: $ \sum_\limits{l,k=0}^{\infty} \binom{l}{k} (-1)^kr^{k-2l} $ .

I feel like I first have to establish absolute convergence for a certain range of values of $|r|$ but I'm not sure how to do that.

Next I think I can then use a double summation like so: $ \sum_\limits{l=0}^{\infty}\sum_\limits{k=0}^{\infty} \binom{l}{k} (-1)^kr^{k-2l} = \sum_\limits{l=0}^{\infty}\sum_\limits{k=0}^{l} \binom{l}{k} (-1)^kr^{k-2l} $ but I'm not sure how to evaluate the sum from that point on.

Answer 1

Considering $$S=\sum_{l=0}^{\infty}\sum_{k=0}^{\infty} \binom{l}{k} (-1)^kr^{k-2l} =\sum_{l=0}^{\infty}r^{-2l}\left(\sum_{k=0}^{\infty} \binom{l}{k} (-r)^k \right)=\sum_{l=0}^{\infty}{r^{-2l}}{(1-r)^l}$$ $$S=\sum_{l=0}^{\infty}\left(\frac{1-r}{r^2}\right)^l=\frac{r^2}{r^2+r-1}$$