Suppose that $\displaystyle f(x) = \sum_{n=0}^\infty a_n x^n$ and that $\displaystyle g(x) = \sum_{n=0}^\infty b_n x^n$. Then 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.