$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $
$$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of partitions of }n}{\text{into an odd number of parts}}$$
Find the truncation of $A (x) B(x)$ to degree $10$.
This is what I have so far:
$$A (x) = \frac{1}{(1-x)(1-x^2)(1-x^3)(1-x^4)\cdots}$$
I believe $B(x) = \frac{1}{(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots}$
$$A(x)B(x) = \frac{1}{(1+x)(1-x)(1+x^2)(1-x^2)(1+x^3)(1-x^3)\cdots}$$
Currently $A(x)B(x)$ is a power representation. My goal is to turn $A(x)B(x)$ into a power series (where it is infinite), then find the polynomial where it goes up to degree $10$.
My question is: 1) Is what I have currently correct?
and
2) How would I use geometric sums to compute the power series expansion of $A(x)*B(x)$? I actually asked this here: text