Infinite Sequences and Series: Formulas for the Remainder Term in Taylor Series

In this video I expand upon my previous video on Taylor and Maclaurin series by going over a few alternative forms of the Remainder Term. Recall that the Remainder Term is just that, a remainder term, and is what is left over when approximating a function with its Taylor series. In the previous video I showed that we can use Taylor’s Inequality to determine the remainder term, but in this video, I show that we can also use an integral form as well as a form called Lagrange’s form of the remainder term. I prove both of these forms of the remainder as well as going over some examples to compare with the previous video’s use of Taylor’s Inequality.

The topics covered in this video are listed below with their time stamps.

- @ 0:00 - Intro
1. @ 2:36 - Recap on Taylor and Maclaurin Series
2. @ 8:26 - Formulas for the Remainder Term in Taylor Series
3. @ 11:32 - Theorem 1: Integral Form of the Remainder Term
- @ 40:48 - Recap of Example 4 from Previous Video
- @ 43:44 - Example 1: sin x
- @ 52:24 - Absolute Value Integral Property
4. @ 1:08:42 - Lagrange's Form of the Remainder Term
- @ 1:13:38 - Theorem 2: Weighted Mean Value Theorem for Integrals
- @ 1:28:20 - Theorem 3: Lagrange's Form of the Remainder Term
- @ 1:40:36 - Example 2: sin x
- @ 1:44:35 - Example 3: x^(1/3)

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