Review Question 10: Taylor and Maclaurin Series

In this video I recap on Taylor polynomials, Taylor series, Maclaurin series, and Taylor's inequality. A Taylor polynomial is a finite sum with terms consisting of the i-th derivative at a divided by the i-th factorial and multiplied by (x - a) to the power of i. A Taylor series is just an infinite Taylor polynomial. A Maclaurin series is a special type of Taylor series where a = 0. A function is equal to its Taylor series if the remainder approaches 0 as n approaches infinity. Taylor's Inequality states that if the (n + 1) derivative of f(x) is less than or equal to a number M then the absolute value of the n-th remainder of a Taylor series is less than M divided by (n + 1) factorial multiplied by the absolute value of (x - a) to the power of (n + 1). Taylor and Maclaurin series are very useful since they can turn complicated functions into a simple sum of many easily calculatable terms, and is how most calculators work behind the scenes!

The timestamps of key parts of the video are listed below:

- Question 10: 0:00
- (a) Taylor polynomial: 0:42
- (b) Taylor series: 2:20
- (c) Maclaurin series: 3:33
- (d) A function equals its Taylor series if the remainder approaches 0: 4:26
- (e) Taylor's Inequality: 5:45

This video was taken from my earlier video listed below:

- Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L

Related Videos:

Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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