Understanding Confusing Expansion: Taylor Series Expansion

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Discussion Overview

The discussion revolves around the Taylor series expansion of a mathematical expression involving a small parameter, ##\beta##. Participants are examining the accuracy of a specific expansion and the steps involved in deriving it, with a focus on potential errors in the derivation process.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on the Taylor expansion of the expression, noting that their approach does not yield the expected result.
  • Another participant confirms the use of the Taylor expansion for ##1/(1-x)## but suggests that it leads to a different outcome than anticipated.
  • A participant emphasizes the importance of considering the ##x^2## term in the expansion, indicating that it may affect the results.
  • One participant expresses a belief that the derived expression is sufficient for their purposes, despite potential inaccuracies in earlier steps.
  • Another participant reports obtaining a different coefficient in the expansion, suggesting that the original source may contain an error, particularly regarding the quadratic term.
  • A later reply identifies a mistake in the expansion process, asserting that a term of ##\beta \hbar \omega## should be present instead of ##\beta \hbar \omega/2##, while agreeing that the final coefficient is correct.
  • Another participant proposes that a typo may have occurred in the original work, which could explain the discrepancies noted in the derivation.

Areas of Agreement / Disagreement

Participants express differing views on the correctness of the expansion and the presence of errors in the original derivation. There is no consensus on the accuracy of the coefficients or the steps taken in the expansion process.

Contextual Notes

Participants note potential limitations in the derivation, including the dependence on the correct handling of terms in the Taylor expansion and the possibility of errors in the original source material.

WWCY
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I came across the following working in my notes and would like some help understanding how the step was done. Many thanks in advance!

The following is the working, and we assume that ##\beta## is small
$$\frac{1}{1+ \beta \hbar \omega /2 + (7/12)(\beta \hbar \omega)^2 +...} \approx 1 - (\beta \hbar \omega)/2 + (5/12) (\beta \hbar \omega)^2 +...$$

I thought of trying the Taylor expansion for ##1/(1-x)## where the ##x## was everything apart from the "1" in the reciprocal term on the LHS but it doesn't seem to work.
 
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WWCY said:
I thought of trying the Taylor expansion for ##1/(1-x)## where the ##x## was everything apart from the "1" in the reciprocal term on the LHS but it doesn't seem to work.
That should be the right procedure, but it does give a slightly different result. Can you give a source for the equation you quoted?
 
## \frac{1}{1-x}=1+x+x^2+x^3+... ##. In this case you need to consider the ## x^2 ## term as well.
 
If we multiply the two expressions, we get ##1 \approx 1+ O(\beta^2)## so my suspicion is, that a) this expression is good enough for what is to come and b) the expression on the right is in some meaning more helpful than the "correct" formula. In short: it suffices and fits the goal.
 
I get a ## -\frac{1}{3} ## instead of the ## +\frac{5}{12} ##. ## \\ ## Edit: I double-checked my arithmetic/algebra, and even worked a numerical example, letting ## \beta \hbar \omega=.02 ##. I think they goofed. It is something they should get right, but textbooks do contain errors on occasion.## \\ ## And you do get a +5/12 on the quadratic term if there is no 2 in the denominator of the ## \beta \hbar \omega ## term. I think it is probably an honest mistake.
 
Last edited:
Thanks everyone for the clarification.

DrClaude said:
That should be the right procedure, but it does give a slightly different result. Can you give a source for the equation you quoted?

It was a solution to one of the problems in Steven Simon's "Oxford Solid State Basics", I'll attach an image below. The step I refer to is the third from bottom.

Screen Shot 2018-08-28 at 6.21.44 PM.png
 

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There is a mistake in step 3: expanding the square in the denominator leads to a term of βħω, not βħω/2. Then the term βħω/2 in step 4 is wrong, but the 5/12 is right.
 
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Looks like the final line is correct=even though they made a couple of errors previous to that.
 
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Just a typo in one line. The author probably copied the previous line, adjusted it and forgot to remove the /2 there.

Another example where more context helps to spot the error.
 
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Cheers guys, I should have been a bit more attentive, and also uploaded the image from the get-go. Apologies.
 
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