Expanding a function for large E using the Taylor Expansion technique

In summary, the conversation was about using Taylor expansion and the speaker was struggling to find any useful results. They asked for hints and explanations, and the other person suggested using a binomial expansion for large values of |E|. The speaker then recognized the binomial and thanked the other person.
  • #1
CricK0es
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Homework Statement
Obtain leading order behaviour of function
Relevant Equations
*See attached image*
I have been playing around with Taylor expansion to see if I can get anything out but nothing is jumping out at me. So any hints, suggestions and preferably explanations would be greatly appreciated as I’ve spent so so long messing around with it and I need to move on...

But as always, thank you
 

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  • #2
We have [tex]
\left(C + \frac{D}{E}\right)^{-1} = \frac{1}{C} \left(1 + \frac{D}{CE}\right)^{-1}.[/tex] For sufficiently large [itex]|E|[/itex] we have [itex]|D/(CE)| < 1[/itex] so we can use a binomial expansion.
 
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  • #3
Ahhhhhh Binomial! Okay. Always seems to be simple things that I don’t recognise that hold me up... pffh Nevermind.

Thank you!
 

FAQ: Expanding a function for large E using the Taylor Expansion technique

What is the Taylor Expansion technique?

The Taylor Expansion technique is a mathematical method used to approximate a function by expressing it as an infinite sum of terms involving the function's derivatives at a single point. This allows for the evaluation of a function at a point using only a finite number of terms.

Why is the Taylor Expansion technique useful for expanding functions for large E?

The Taylor Expansion technique is useful for expanding functions for large E because it allows for the approximation of a function in terms of its derivatives at a single point, making it easier to evaluate the function at large values of E. This can be especially useful in scientific and engineering applications where functions with large values of E are often encountered.

What is the difference between the Taylor Series and Taylor Expansion?

The Taylor Series and Taylor Expansion are often used interchangeably, but they have a subtle difference. The Taylor Series is the infinite sum of terms used to approximate a function, while the Taylor Expansion is the process of using a finite number of terms from the Taylor Series to approximate the function at a specific point.

How does the number of terms in the Taylor Expansion affect its accuracy?

The number of terms used in the Taylor Expansion directly affects its accuracy. As more terms are included, the approximation becomes more accurate. However, including too many terms can also lead to computational errors. It is important to find a balance between accuracy and computational efficiency when choosing the number of terms for a Taylor Expansion.

Can the Taylor Expansion be used for any function?

No, the Taylor Expansion can only be used for functions that are infinitely differentiable, meaning that they have an infinite number of continuous derivatives. If a function is not infinitely differentiable, then the Taylor Expansion cannot be used to approximate it.

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