How to understand Taylor/Mclaurin series?

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SUMMARY

This discussion focuses on understanding Taylor and Maclaurin series, crucial concepts in Calculus 2. Maclaurin series are a specific case of Taylor series, where the expansion occurs around the point a = 0. Participants recommend resources such as Paul's online notes and Khan Academy videos for clearer explanations. Additionally, the discussion emphasizes the importance of finding the constants C in the Taylor series representation of functions.

PREREQUISITES
  • Understanding of basic calculus concepts, including derivatives and integrals.
  • Familiarity with series expansions, specifically Maclaurin series.
  • Knowledge of function manipulation techniques, such as rewriting functions in terms of 1/(1-x).
  • Ability to interpret mathematical notation related to series and summations.
NEXT STEPS
  • Explore Taylor series derivation methods and practice finding constants C.
  • Review Paul's online notes on Taylor series for comprehensive explanations.
  • Watch Khan Academy videos on Taylor series for visual and conceptual understanding.
  • Investigate 3Blue1Brown's videos for intuitive insights into series expansions.
USEFUL FOR

Students in Calculus 2, educators teaching calculus concepts, and anyone seeking to deepen their understanding of series expansions in mathematics.

Jrohazn
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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.
 
Khan Academy has a few well put together videos on Taylor Series.

Here are two




Hope this helps and good luck on your final.
 
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.
 
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.
 

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