Why does this equation holds?

Question

Could anyone tell me why following equation holds?

$ \sum_{n \geq 0} x^n \sum_{i \geq 0} \binom{i}{n-i} = \sum_{i \geq 0} x^i \sum_{n \geq 0} \binom{i}{n-i} x^i$

Answer 1

I'm not sure if it does.

For example, in the right hand side the coefficient of $x$ is $$\binom{0}{1}+\binom{1}{0}=1,$$ whereas on the left hand side the coefficient of $x$ is $0$..

But maybe i'm missing out on something...