Expanding Taylor Series to Get Approximate Answer

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Discussion Overview

The discussion revolves around the expansion of Taylor series to approximate expressions in a mathematical model. Participants explore how to linearize a given equation involving exponential functions and consider the implications of different forms of the equations for Taylor series expansion.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant presents an expression involving an exponential function and seeks clarification on how it was approximated using a Taylor series.
  • Another participant reiterates the expression and provides the first two terms of the series expansion for the exponential function, indicating the approximation used.
  • A participant notes that for small values of \(x\), the exponential function can be approximated as \(e^{x} \approx 1 + x\), suggesting a general principle of Taylor expansion.
  • One participant questions whether to express a different equation in the form \((K+v_t)g\) before expanding \(g\) in a Taylor series, or to expand the Taylor series of the entire expression directly.
  • Another participant echoes the previous question about the appropriate form for expansion, indicating uncertainty about the method to apply.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the best approach to expand the Taylor series for different forms of the equations. There is no consensus on the method to apply for the new expression involving \(K\) and \(v_t\).

Contextual Notes

Participants do not clarify assumptions about the size of variables or the context in which the approximations are valid. The discussion remains focused on the mechanics of Taylor series without resolving the specific application to the new equation.

Dustinsfl
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$1+v_{t+1} = (1+v_t)\exp\left(-rv_{t-1}\right)\approx (1+v_t)(1-rv_{t-1})$

The book is linearizing the model where we generally use a Taylor Series.

How was the expression expanded in the Taylor Series to get the approximate answer?

Thanks.
 
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dwsmith said:
$1+v_{t+1} = (1+v_t)\exp\left(-rv_{t-1}\right)\approx (1+v_t)(1-rv_{t-1})$

The book is linearizing the model where we generally use a Taylor Series.

How was the expression expanded in the Taylor Series to get the approximate answer?

Thanks.

The second set of brackets contains the first two terms of the series expansion of

\[ \exp(-rv_{t-1}) =\sum_{k=0}^{\infty}\frac{(-rv_{t-1})^k}{k!}= 1-rv_{t-1}+{\text{O}}((rv_{t-1})^2)\]

CB
 
If $x$ is small, than $e^{x} \approx 1 + x$.

Edit - too slow :-)
 
CaptainBlack said:
The second set of brackets contains the first two terms of the series expansion of

\[ \exp(-rv_{t-1}) =\sum_{k=0}^{\infty}\frac{(-rv_{t-1})^k}{k!}= 1-rv_{t-1}+{\text{O}}((rv_{t-1})^2)\]

CB

Now if i was dealing $K+v_{t+1}=(K+v_t)\left[1 + r \left(1- \frac{K+v_t}{K}\right)\right]$, would I want to put it in the form $(K+v_t)g$ and then expand g in the Taylor series to the linear terms?
 
dwsmith said:
Now if i was dealing $K+v_{t+1}=(K+v_t)\left[1 + r \left(1- \frac{K+v_t}{K}\right)\right]$, would I want to put it in the form $(K+v_t)g$ and then expand g in the Taylor series to the linear terms?

Or do we expand the Taylor Series of

$$
\left[1 + r \left(1- \frac{K+v_t}{K}\right)\right]
$$

with the \(K+v_t\) inside?
 
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