# Taylor and MacLaurin Series

A series of free online calculus lectures and solutions. The lessons here look at the Taylor and MacLaurin Series.

The following diagrams show the Taylor Series and some examples of the MacLaurin Series. Scroll down the page for more examples and solutions using the Taylor Series and MacLaurin Series.

Taylor and Maclaurin Series - Example 1
An example of finding the Maclaurin series for a function is shown.

Taylor and Maclaurin Series - Example 2
A complete example of finding a Taylor series for the function ln(x) centered at a = 2 is shown.

Using Maclaurin/Taylor Series to Approximate a Definite Integral to a Desired Accuracy
This video uses Maclaurin/Taylor series and the Alternating Series Estimation Theorem to approximate a definite integral to within a desired accuracy.

Taylor’s Inequality
Examples are shown using Taylor’s Inequality. The first part shows that a series expansion is valid using Taylor’s Inequality. The second part shows how to use Taylor’s Inequality to estimate how accurate a Taylor Polynomial will be.

Taylor / Maclaurin Series Expansion - Proof of the Formula
Taylor / Maclaurin Series Expansion - Deriving the Formula. In this video, I show how to find the Taylor series expansion for a function, assuming that one exists! It is nothing too heavy: we just take derivatives and plug in the value at which we are centering the function. I do not find any Taylor series of specific functions in this video, nor do I justify when a Taylor series expansion is valid (not all functions have power series expansion!).

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.