I am unable to think about how to prove this result related to generating functions.
Let $\{a_n\}_{n=0}^\infty$ be an infinite sequence and $b_n = \sum_{i=0}^n a_i $. Let $G_a(x)$ be the generating function for $\{a_n\}$ and $G_b(x)$ be the generating function for $\{b_n\}$. Prove that $G_a(x) = (1-x) G_b(x)$ .
Can someone please tell how to prove this result.
HINT
change the order of summations $$ G_b(x) = \sum_{n=0}^\infty b_n x^n = \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i\right) x^n = \sum_{i=0}^\infty a_i \left(\sum_{n=i}^\infty x^n\right) = \sum_{i=0}^\infty a_i x^i \left(\sum_{n=0}^\infty x^n\right) $$ Can you finish it now?