Multiplying a finite and infinite series

Question

I am struggling to find the product of these two series:

$(\sum_{i=0}^np_ix^i)(\sum_{i=0}^\infty (-1)^ix^i)$.

I thought the solution was $\sum_{i=0}^\infty[p_0(-1)^i+p_1(-1)^{i-1}+\cdots+p_n(-1)^{i-n}+(-1)^{i-(n+1)}+\cdots+(-1)^0]x^i$ where $0\leq n<\infty$ but this does not seem to be right.

I calculated the product in MAPLE to see what the first 5 terms was, and I got the following: $$ p_0+(p_1-p_0)x+(p_2-p_1+p_0)x^2+(p_3-p_2+p_1-p_0)x^3+(p_4-p_3+p_2-p_1+p_0)x^4+(p_5-p_4+p_3-p_2+p_1-p_0)x^5 $$

So I guess I am struggling with putting this in the general sense. Any help would be much appreciated. Thanks.

Answer 1

For each $n$, write $$c_n=\sum_{i=0}^np_ix^i=p_0+p_1x+p_2x^2+p_3x^3+\dots+p_nx^n.$$ Then $$\begin{align} \sum_{i=0}^np_ix^i\cdot\left( \sum_{i=0}^{\infty}(-1)^ix^i\right)&=c_n\cdot (1-x+x^2-x^3+\dots)\\ &=c_n-c_nx+c_nx^2-c_nx^3+\dots\\ \end{align}$$ From here, simplify and combine like terms. Hope this help.