The derivative of a Taylor series?

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Discussion Overview

The discussion revolves around the concept of differentiating a Taylor series, including the mathematical principles involved and the participants' varying levels of familiarity with the topic. It touches on theoretical aspects of calculus and the application of Taylor series in differentiation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant expresses confusion about the derivative of a Taylor series, indicating a lack of prior exposure to the concept.
  • Another participant explains that if f(x)=Ʃanxn in some interval, then f'(x)=Ʃnanxn-1 in the interior of that interval, citing uniform convergence as a reason for termwise differentiation.
  • A later reply suggests that the differentiation of the Taylor series resembles the differentiation of finite polynomials.
  • One participant expresses difficulty with sigma notation and seeks clarification on the derivative of a specific Taylor series expansion.
  • Another participant confirms that differentiating each term individually yields the same result, provided the series converges.
  • One participant suggests using the power rule for differentiation.

Areas of Agreement / Disagreement

Participants generally share insights on the differentiation of Taylor series, but there is no consensus on the level of difficulty or the clarity of the explanation, as some express confusion while others provide technical details.

Contextual Notes

Some participants indicate uncertainty regarding the application of convergence conditions and the implications of differentiating series termwise.

Who May Find This Useful

Students learning calculus, particularly those interested in series and differentiation, as well as educators looking for insights into common student misconceptions about Taylor series.

badamann
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I took my first calculus class over the last two semesters, and my teacher and I privately worked on some harder material together. Toward the end of the school year he gave me a question that I never answered and never found an answer for. It asked me to find the derivative of a Taylor series. I don't remember the specific problem, but if someone could shed some light, that'd be lovely!
 
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you are kidding right? i.e. what do you think?
 
I'm not kidding at all. At the time I received the problem I'd never heard of a Taylor series, and I haven't seen a problem with one since. It's something I've never had the opportunity to learn at school thus far.
 
I'm not sure if this is exactly what you're looking for, but if f(x)=Ʃanxn in some interval, then f'(x)=Ʃnanxn-1 in the interior of that interval; that is, you can differentiate the series termwise. This is because of uniform convergence, which can be shown with the Weierstrass M-test.
 
Really, it's a very plausible generalization of what you already know for finite polynomials.
 
I'm really just caught up in looking at the sigma notation.. It throws me out of whack. I'm assuming I'm making a much bigger deal out of this than I should be.

This may be a better question:

If f(x) = f(a) + [f'(a)/1!](x-a) + [f"(a)/2!](x-a)^2 + ...
Then f'(x) = ?
 
Last edited:
badamann said:
This may be a better question:

If f(x) = f(a) + [f'(a)/1!](x-a) + [f"(a)/2!](x-a)^2 + ...
Then f'(x) = ?

Exactly what you would get if you differentiated each term individually.
Provided it converges.
 
badamann, just use the power rule.
 
Thank you. I'm not sure why I made it out to be quite so difficult... :(
 

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