Multiplying power series I'm trying to understand how the expression
$$
\sum\limits_{n=0}^\infty x^n \cdot \sum\limits_{n=0}^\infty x^n
$$
results in the following expression:
$$
\sum\limits_{n=0}^\infty (1+n)x^n.
$$
Could you please walk me through it?

A simple observation might be enlightening:
$$(1+x+x^2+x^3)(1+x+x^2+x^3)\\
=1+x+x^2+x^3+x+x^2+x^3+x^4+x^2+x^3+x^4+x^5+x^3+x^4+x^5+x^6\\
=1+2x+3x^2+4x^3+3x^4+2x^5+x^6.$$
