Suppose that $\displaystyle f(x) = \sum_{n=0}^\infty a_n x^n$
and that $\displaystyle g(x) = \sum_{n=0}^\infty b_n x^n$.
We can get the power series for $f(x)+g(x)$, $f(x)g(x)$ and
$f(x)/g(x)$ by adding, multiplying, and dividing these
expressions, as if they were polynomials:

Computing $f(x)/g(x)$, however, is trickier as we have to
perform long division, and treat larger powers of $x$ as being
less important than smaller powers. These ideas are explained in
the following video. Unless you are told otherwise by your
instructor, you will at the most be working with the sums and
differences of series, and not the more complicated products and
quotients.