How to understand Taylor/Mclaurin series?

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In summary, the video provides an overview of Taylor series, explains how to calculate the constants in the series, and provides a proof of the series.
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I'm in Calculus 2, and I have a final coming up. I did extremely well in all other sections, but this section is extremely confusing. I can represent functions into a basic MacLaurin series, and I can also take the derivative of a function and find the series through manipulation of the function into 1/1-x and take the integral of that. Taylor series, though, is EXTREMELY confusing. I have no idea what to do, and no matter how hard I try, I can't find the pattern. Please help!
 
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Jrohazn said:
I'm in Calculus 2, and I have a final coming up. I did extremely well in all other sections, but this section is extremely confusing. I can represent functions into a basic MacLaurin series, and I can also take the derivative of a function and find the series through manipulation of the function into 1/1-x and take the integral of that. Taylor series, though, is EXTREMELY confusing. I have no idea what to do, and no matter how hard I try, I can't find the pattern. Please help!
Maclaurin series are just a special case of Taylor series, expanding the function about the value 0 instead of the more general expansion about a point x=a.
Rather than launch into an explanation of Taylor series from scratch, how about you find a reasonable online explanation and come back here to ask about any parts you don't understand? Paul's online notes are generally very good: http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx

Edit: well, that's an endorsement - we independently picked the same site.
 
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Another endorsement from me for Paul as a reference for all things calculus.
 
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Khan Academy has a few well put together videos on Taylor Series.

Here are two




Hope this helps and good luck on your final.
 
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this video also is great
 
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MAGNIBORO said:
this video also is great

Knowing the quality of 3Blue1Brown's videos I would certainly recommend that OP check that video out.
 
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Yes that is a good video. I will take a look at some of other 3Blue1Brown videos.
 
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Taylor series is just saying "can I rewrite a function as an infinite summation of C(x-a)^(increasing powers)".

So say I have the function 2x. Taylor series asks, can 2x be written as C1(x-a)0 + C2(x-a)1 +C3(x-a)2 +C4(x-a)3 + ...

The answer is yes, and you can do it for any function. To show this, you have to find out a method to calculate the constants C.

Here is a great and simple proof of Taylor Series.
 

FAQ: How to understand Taylor/Mclaurin series?

1. What is a Taylor/Mclaurin series?

A Taylor/Mclaurin series is a representation of a function as an infinite sum of terms, where each term is a polynomial expression of increasing degree. It is used to approximate the value of a function at a certain point by using the function's derivatives at that point.

2. Why are Taylor/Mclaurin series important?

Taylor/Mclaurin series are important because they provide a way to approximate and analyze complex functions, especially in calculus and physics. They can also be used to solve differential equations and model real-life phenomena.

3. How do I find the Taylor/Mclaurin series for a given function?

To find the Taylor/Mclaurin series for a function, you can use the Taylor/Mclaurin series formula, which involves taking derivatives of the function at a specific point and plugging them into the formula. Alternatively, you can use the Maclaurin series method, which involves finding a pattern in the derivatives and using it to write a general formula for the series.

4. What is the difference between Taylor and Mclaurin series?

The main difference between Taylor and Mclaurin series is the point at which they are expanded. Taylor series are expanded around a specific point, while Mclaurin series are expanded around the origin (x=0). In other words, Mclaurin series are a special case of Taylor series.

5. How accurate are Taylor/Mclaurin series?

The accuracy of Taylor/Mclaurin series depends on the function and the number of terms used in the series. Generally, the more terms included in the series, the more accurate the approximation will be. However, Taylor/Mclaurin series only provide an approximation and may not give the exact value of a function at a specific point.

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