Operations on Maclaurin series including term-by-term differentiation and integration, adding and subtracting coefficients, multiplying and dividing series to find initial terms, using substitutions for composite functions, and applying series expansions to evaluate limits.
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+a1x+a2x2+... 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.
Nice work completing Operations on Maclaurin Series, here's a quick recap of what we covered:
Exercises checked off
Operations on Maclaurin series including term-by-term differentiation and integration, adding and subtracting coefficients, multiplying and dividing series to find initial terms, using substitutions for composite functions, and applying series expansions to evaluate limits.
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+a1x+a2x2+... 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.
Nice work completing Operations on Maclaurin Series, here's a quick recap of what we covered:
Exercises checked off