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 nth 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.