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Infinite Sequences and Series: Taylor and Maclaurin Series

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First published at 06:02 UTC on February 7th, 2020.

#mathematics
#calculus
#science

Math Easy Solutions

MES

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In this video I go over further into the wonderful world of sequences and series, and this time go over Taylor and Maclaurin series. These series are precise and ingenious ways of approximating many functions by only knowing all of the function’s derivatives at a specific value; in the case of the Maclaurin series it is as x = 0. These series can also be used to integrate functions that don’t have elementary antiderivatives and thus are utilized in many computers and calculators as the basis behind many types of calculations.

Interestingly, it was this very concept of Taylor and Maclaurin series that first gave me the idea of creating a YouTube channel to better learn and organize my learning of mathematics and beyond. Taylor and Maclaurin series, as well as infinite series in general, require the coming together of many mathematical concepts and thus I had then realized while still in University that I need to fully refine my mathematical knowledge. And this particular video is the perfect illustration of such a culmination of so many various mathematical concepts.

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

@ 1:14 – MES Note on Taylor and Maclaurin Series

1. @ 5:16 - Introduction to Taylor and Maclaurin Series
- @ 21:38 - Theorem 1
- @ 22:41 - Taylor Series
- @ 24:43 - Maclaurin Series
- @ 26:25 - Historical Note on Taylor and Maclaurin
- @ 27:37 - Example 1
2. @ 35:00 - When is a Function Equal to the Sum of Its Taylor Series?
- @ 36:18 - Taylor Polynomials
- @ 49:01 - Theorem 2
- @ 49:49 - Taylor's Inequality
- @ 51:36 - Proof for n = 1
- @ 1:14:09 - Equation 1
- @ 1:15:03 - Note on Alternatives to Taylor's Inequality
- @ 1:17:41 - Example 2: e^x
- @ 1:30:11 - Historical Note on Calculating the Number e
- @ 1:31:15 - Example 3
- @ 1:35:04 - Example 4: sin x
- @ 1:48:33 - Example 5: cos x
- @ 1:55:31 - Historical Note on Maclaurin Series
- @ 1:56:23 - Example 6: x cos x
- @ 1:58:04 - Example 7
- @ 2:11:23 - Note on Power Series
3. @ 2:12:30 - Comparing Taylor and Maclaurin Series
4. @ 2:15:44 - Example 8: (1 + x)^k
- @ 2:32:26 - The Binomial Series
- @ 2:45:43 - Example 9
5. @ 2:59:09 - Table 1: Important Maclaurin Series and Their Radii of Convergence
6. @ 3:04:12 - Integrating Functions that Don't Have Elementary Antiderivatives
- @ 3:09:35 - Example 10: e^(-x^2)
7. @ 3:28:43 - Example 11: Evaluating Limits
8. @ 3:38:26 - Multiplication and Division of Power Series
- @ 3:39:33 - Example 12
9. Exercises
- @ 3:57:28 - Exercise 1: A Function Not Equal to its Maclaurin Series
- @ 4:17:05 - Exercise 2: Proof of Taylor's Inequality for n = 2
- @ 4:37:01 - Exercise 3: Proof of Binomial Series