Taylor Approximation Proof for P(r) using Series Expansion

In summary, the problem involves proving that the function P(r) is approximately equal to 3r^3/4a^4 using Taylor series expansion. This can be done by constructing the Taylor series and neglecting higher order terms, or by expanding the exponential and terminating at a certain point. The solution involves calculating derivatives and expanding to the fourth order.
  • #1
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[SOLVED] Taylor approximation

Homework Statement


I have an exact funktion given as:

[tex]P(r)=1-e^{\frac{-2r}{a}}(1+\frac{2r}{a}+\frac{2r^2}{a^2})[/tex]

I need to prove, by making a tayler series expansion, that:
[tex]P(r)\approx \frac{3r^3}{4a^4}[/tex]

When [tex]r \prec \prec a[/tex]


The Attempt at a Solution


I am lost when it comes to these Taylor approximations. It should be a fairly easy problem, but don't know how to handle it.
Some help on how to do this would be appreciated.
 
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  • #2
In general, the Taylor series of a function f(r) around 0 is given by
[tex]f(r) = \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) r^n = f(0) + f'(0) r + 1/2 f''(0) r^2 + 1/6 f'''(0) r^3 + 1/24 f''''(0) r^4 + \cdots[/tex]
where [itex]f^{(n)}[/itex] denotes the nth derivative. Usually, one stops the expansion after a certain number of terms (e.g. [itex]f(r) \approx f(0)[/itex] is the "first order expansion", including the first derivative term gives the "second order expansion", etc.*)

So basically, all you need to do for this problem is construct the Taylor series (you'll need to calculate the derivatives) and terminate it at a proper point. Or alternatively, if you know the expansion of the exponential, you can plug it in, expand the brackets, and again terminate it somewhere (that is, neglect all powers [itex]r^n, r^{n+1}, r^{n+2}[/itex], etc. for an nth order expansion).

*) I also see people calling f(0) the zero'th order expansion, f'(0) r the first order, etc. Sometimes that is confusing but so be it
 
  • #3
Thanks for the quick reply

By expanding the exponential to fourth order I was able to find the desired result
 

1. What is the Taylor Approximation Proof for P(r) using Series Expansion?

The Taylor Approximation Proof for P(r) using Series Expansion is a mathematical technique used to approximate the value of a function, P(r), at a given point, r. It involves expressing the function as a polynomial, or series, of infinitely many terms and using a finite number of terms to approximate the value of the function at a specific point.

2. Why is Taylor Approximation important in scientific research?

Taylor Approximation is important in scientific research because it allows us to approximate complex mathematical functions using simpler polynomials. This makes it easier to perform calculations and make predictions, especially in situations where the exact value of a function is difficult or impossible to determine.

3. How is the Taylor Approximation Proof for P(r) derived?

The Taylor Approximation Proof for P(r) is derived using the concept of a power series, which is a mathematical series where each term is a constant multiplied by a variable raised to an increasing power. By taking the derivative of this power series, we can determine the coefficients of the polynomial used in the Taylor Approximation.

4. What is the relationship between Taylor Approximation and calculus?

Taylor Approximation is closely related to calculus, as it involves taking derivatives of a function to determine the coefficients of the polynomial used in the approximation. It is also used in calculus to approximate a function at a specific point, which can then be used to calculate limits, derivatives, and integrals.

5. Can Taylor Approximation be used for any type of function?

Yes, Taylor Approximation can be used for any type of function as long as it meets certain conditions, such as being continuous and having continuous derivatives of all orders. However, the accuracy of the approximation may vary depending on the complexity of the function and the number of terms used in the polynomial.

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