Taylor Series: What Is the Significance of the a?

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Discussion Overview

The discussion revolves around the significance of the variable "a" in Taylor series expansions, particularly its role as the center of the series and how this choice affects the series' representation and convergence properties. Participants explore both the mathematical implications and the physical interpretations of this variable.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the variable "a" and its significance in the context of Taylor series, suggesting that clarity is needed regarding its definition.
  • One participant notes that the choice of "a" as the center of the series affects the convergence and representation of the function, indicating that different choices can lead to different series expansions.
  • Another participant mentions that if the series converges everywhere, the Taylor series remains consistent, but the approximation may vary depending on how close one is to "a".
  • There is a humorous exchange about the arbitrary nature of variable names, with some participants joking about the common use of "a" versus other notations like "x_0" or "c".
  • One participant emphasizes that around the point "a", functions can be approximated by polynomial expressions, which are simpler to handle than other types of functions.

Areas of Agreement / Disagreement

Participants generally agree that "a" is the center of the Taylor series, but there is no consensus on the best way to denote it or the implications of its choice on the series' behavior. The discussion includes both technical explanations and light-hearted banter, indicating a mix of serious inquiry and camaraderie.

Contextual Notes

Some assumptions about the nature of convergence and the behavior of functions near "a" are not fully explored, leaving room for further discussion on these mathematical properties.

abdo799
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i watched a lot of videos and read a lot on how to choose it, but i what i can't find anywhere is, what's the physical significance of the a, if we were to draw the series, how will the choice of a affect it?
 
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What a ? We at PF are bad at telepathy, so we need an example or something to guess what you mean with a...
 
Variable names are arbitrary. You can call everything you want "a".

Sometimes it is possible to guess what is meant if everyone uses the same variable names, but that is not the case here.
 
:smile: actually, in a flash of telepathic insight ( and googling 'taylor series' :rolleyes: ) I'm almost sure what abdo means, but rewarding bad practice isn't good practice ...
 
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Probably, Abdo means the ##a## in the representation ##f(x)=\sum_{n=0}^\infty c_n(x-a)^n##. This ##a## is simply the point about which the expansion is taken. The sum will be different, and will perhaps not converge, if we change this point.

For example, expanding ##e^x## about ##0## gives ##e^x=\sum_{n=0}^{\infty}x^n/n!## but expanding about ##1## gives ##e^x=\sum_{n=0}^{\infty}e(x-1)^n/n!##
 
he perhaps means this "a":
 
hahahahhahaha, sry, it's just that everybody calls the center of the series a, nice video though :D
 
u see, this is what i love about PF, everybody is friendly enough to joke about it if i made a mistake instead of keep reporting me or removing the thread
 
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If the series converges to the function everywhere ("infinite" radius of convergence), then the full taylor series is the same everywhere, although the terms look differently - so if you just take a few of them, the approximation will look differently. Typically the approximation is good close to "a".

If the series does not converge to the function everywhere, it can look completely different for different "a".
 
  • #10
Without a this the Tylor series and we don't have informations on it ... :biggrin:

ok, it is a stupid joke. As said before I agree assuming ##a## the center of the Taylor series. Physical interpretation is that around ##a## you can approximate, well as you want, your function by a polynomial expressions .. This is good because polynomials are simple to treat instead other functions ...
 
  • #11
abdo799 said:
hahahahhahaha, sry, it's just that everybody calls the center of the series a, nice video though :D
I denote it by ##x_0## or when lazy ##c## for centre. So you are wrong, not everybody calls it ##a##.
 
  • #12
wrobel said:
he perhaps means this "a":


I don't get it. What "a"?
 
  • #13
the aria of the black haired guy consists only of "a-a-a-a-a"
 
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