Let $\sum a_n$ and $\sum b_n$ be two absolutely convergent series. My question is
Is it possible to take the product of the two series $(\sum a_n)(\sum b_n)$ and get a result different from the Cauchy product? In other words, is the product of two series well-defined? What if $\sum a_n$ and $\sum b_n$ are conditionally convergent?
My motivation for the question is as follows. Cauchy says for two series $\sum a_n$ and $\sum b_n$, we should write
\begin{align*} \left(\sum a_nx^n\right)\left(\sum b_nx^n\right) &= (a_0 + a_1 x + a_2 x^2 + \cdots)(b_0 + b_1 x + b_2 x^2 + \cdots) \\ &= a_0b_0 + (a_0b_1 + a_1b_0)x + (a_0b_2 + a_1b_1 + a_2b_0)x^2 + \cdots \\ &= \sum_{n=0}^\infty \left(\sum_{k=0}^n a_kb_{n-k}\right) x^n \end{align*} and set $x = 1$ to get the desired formula.
But this does some sneaky re-arrangements (I believe). If both $\sum a_n$ and $\sum b_n$ are absolutely convergent, I think we should have no problems in re-arrangements, but I am not sure. So why can't we write
\begin{align*} \left(\sum a_nx^n\right)\left(\sum b_nx^n\right) &= (a_0 + a_1 x + a_2 x^2 + \cdots)(b_0 + b_1 x + b_2 x^2 + \cdots) \\ &= a_0(b_0 + b_1 x + b_2 x^2 + \cdots) + a_1x(b_0 + b_1 x + b_2 x^2 + \cdots) + \cdots \\ &= a_0 \left(\sum b_n\right) + a_1x \left(\sum b_n\right) + \cdots \\ &= \sum_{i=0}^\infty a_ix^i \left(\sum b_nx^n\right) . \end{align*}
Would those lead to the same result?