What is the correct power series for $g(t)$?

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Discussion Overview

The discussion revolves around determining the correct power series representation for the function \( g(t) \). Participants explore different formulations and definitions related to power series, particularly focusing on the Taylor series expansion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes a power series for \( g(t) \) as \( g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!} \), questioning its validity.
  • Another participant expresses skepticism about the proposed recursive definition, suggesting it may not be a suitable approach.
  • A third participant reiterates the same power series formulation and asserts that, as stated, it only represents the zero function.
  • A fourth participant provides the Taylor series expansion for \( g(t) \) around a point \( t_0 \), clarifying the role of derivatives in the series.
  • A later reply acknowledges a misunderstanding in the original question, indicating a realization of an error in their reading.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct formulation of the power series for \( g(t) \). Multiple competing views and interpretations of the series exist, and the discussion remains unresolved.

Contextual Notes

There are indications of misunderstandings regarding the definitions and formulations of power series, particularly in the context of recursive definitions versus Taylor series. The discussion highlights the need for clarity in the definitions used.

Dustinsfl
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What would be the power series of $g(t)$?

$$
g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!}
$$

This?
 
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?? What EXACTLY are you trying to do? I'm guessing the recursive definition you have suggested is not a good way to go.
 
dwsmith said:
What would be the power series of $g(t)$?

$$
g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!}
$$

This?

There is a mistake in your post, as posted there is no such function other than the zero function.

CB
 
The power series of a function around a point \( t_0 \) is $$ g(t) = \sum_{n=0}^{\infty} \frac{(t-t_0)^n g^{(n)}(t_0)}{n!} .$$ Note that the \( g^{(n)}(t_0) \) denotes the derivative evaluated at \( t_0 \), where \( g^{(0)}(t_0) = g(t_0) \).

Does this help you?

Edit: Sorry, to be specific this is the Taylor power series.
 
There was a mistake in what I was reading, i.e. is read it wrong. I see what my issue was so the question I asked was wrong.
 

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