Understanding Power Series Multiplication Step Working on Spivak's Calculus problems, I searched online, trying to understand the solution provided for Problem 4a of Chapter 2. I found the question I needed:
Spivak's Calculus - Exercise 4.a of 2nd chapter.
However, the answer provided there started with the following equation, and then went on from it to explain other things, which I could understand. But this part, which lies at the foundation of the argument, I don't understand.
Could I get an explanation for why it is that:
$$
\begin{align}
\left(\sum_{k=0}^\infty a_kx^k\right)\left(\sum_{k=0}^\infty b_kx^k\right)
&=\sum_{k=0}^\infty\left(\sum_{j=0}^k a_j\color{#C00000}{x^j}b_{k-j}\color{#C00000}{x^{k-j}}\right)
\end{align}
$$
Keep in mind I'm a beginner, working on an introductory Calculus book as my first exposure to the subject. I'd appreciate both intuitive and rigorous answers.

Multiply $$(ax^2+bx^3)(cx^2+dx^3).$$
You use distributive law to open brackets. In what ways can you get a term with $x^5$?
The degrees of the terms you multiply from each factor should add up to $5$. In general you are applying distributive and gathering together terms of degree $5$. So, you multiply together terms from each factor with degrees that add up to $k$. If the first has degree $j$ the second must have degree $k-j$ in order to add up to $k$.
