The Maclaurin series for a composite function ​f(g(x))​ is

Basically, we replace ​xn​ with ​[g(x)]n.

We can add or subtract Maclaurin polynomials by adding or subtracting the coefficients of each power of ​x​ from the original polynomials.

We can integrate a function by integrating its Maclaurin polynomial term by term.

We can differentiate a Maclaurin series using the power rule on each term.

We can find the first few terms in the Maclaurin Series for a product ​f(x)×g(x)​ by multiplying out the first few terms in the Maclaurin Series for ​f​ and ​g.

We can perform "division" of Maclaurin series by assuming that the result of the division is a series with

where ​an​​ are the coefficients of the ​n​th power term. Then, we multiply ​g(x)f(x)​​ and ​a0​+a1​x+a2​x2+...​ by ​g(x). Finally, we solve for the unknown coefficients using the Maclaurin series of ​f(x).

Certain limits may be evaluated with L’Hôpital's rule but require difficult or repeated differentiation. Sometimes, expanding a function within the limit using its Maclaurin series and simplifying is an easier method for evaluating these limits.

This kind of simplification often works because we know that after a certain point, every term will tend to ​0, allowing us to focus on the few terms that do not.