Taylor expansion in infinity?

In summary, the conversation discusses the use of Taylor series for large values of x. It is suggested to calculate the Taylor series of f(1/x) and evaluate it around 0 for good approximations. However, this may not work for divergent series and a Laurent series may be needed. The main goal is to examine the behavior of the given function for large x and approximate it in that case.
  • #1
Vrbic
407
18

Homework Statement


How to use Taylor series for condition x>>1? For example [itex]f(x)=x\sqrt{1+x^2}(2x^2/3-1)+\ln{(x+\sqrt{1+x^2})}[/itex]

Homework Equations

The Attempt at a Solution


I try to derived it and limit to infinity...for example first term [itex]\frac{x^4}{3\sqrt{1+x^2}}[/itex]. Limit this to infinity is obviously infinity. Any advice?
Thank you.
 
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  • #2
Well, the function goes to infinity, so every approximation should do the same.
What exactly do you want to calculate or approximate where?

In general, to get good approximations for large x, you can calculate the taylor series of f(1/x) and evaluate it around 0. That won't work with divergent series, however, where you might need a Laurent series.
 
  • #3
mfb said:
Well, the function goes to infinity, so every approximation should do the same.
What exactly do you want to calculate or approximate where?

In general, to get good approximations for large x, you can calculate the taylor series of f(1/x) and evaluate it around 0. That won't work with divergent series, however, where you might need a Laurent series.
Nice trick, I tried it but for all derivative limit to 0 is divergent...for example 1.derivative of my f(1/x) is [itex] -\frac{1+1/x^2}{\pi x^4}[/itex]. Yes I would like to examine a behaviour of this function for large x and approximate it for this case.
 
  • #4
Well you'll need a Laurent series here, as I mentioned. If it stops, then you have a good approximation.
 
  • #5


I would approach this problem by first understanding the concept of Taylor series and its application in approximating functions. The Taylor series expansion is a powerful tool in mathematics that allows us to approximate a function using a polynomial of infinite degree. This is particularly useful when dealing with functions that are difficult to integrate or differentiate.

In this case, we are dealing with the condition x >> 1, which means that x is much larger than 1. In order to use the Taylor series expansion, we need to choose a point around which we will expand the function. In this case, it would make sense to choose x = infinity.

The Taylor series expansion around x = infinity is given by:

f(x) = f(\infty) + f'(\infty)(x-\infty) + \frac{f''(\infty)}{2!}(x-\infty)^2 + ...

Since we are dealing with the condition x >> 1, we can assume that x-\infty \approx x. Therefore, the above equation can be simplified to:

f(x) \approx f(\infty) + f'(\infty)x + \frac{f''(\infty)}{2!}x^2 + ...

Now, we need to evaluate the derivatives of the function at x = \infty. This can be done by taking the limit as x approaches infinity of the derivatives of the given function. For example, to find the first derivative, we would take the limit as x approaches infinity of the first derivative of the given function:

f'(\infty) = \lim_{x\to\infty} \frac{d}{dx}(x\sqrt{1+x^2}(2x^2/3-1)+\ln{(x+\sqrt{1+x^2})})

Once we have evaluated the derivatives at x = \infty, we can substitute them into the Taylor series expansion and simplify the equation. This will give us an approximation for the given function for the condition x >> 1.

In summary, to use the Taylor series for the condition x >> 1, we need to choose a point around which we will expand the function (in this case, x = \infty), evaluate the derivatives of the function at that point, and substitute them into the Taylor series expansion. This will give us an approximation for the given function for the condition x >> 1.
 

1. What is a Taylor expansion in infinity?

A Taylor expansion in infinity is a mathematical concept that involves representing a function as an infinite sum of terms, each of which is a multiple of a power of the variable. This is done in order to approximate the original function and make calculations easier.

2. How is a Taylor expansion in infinity calculated?

To calculate a Taylor expansion in infinity, you must first determine the coefficients of the terms in the expansion. This can be done using the Taylor series formula, which involves taking derivatives of the function at a specific point. Once the coefficients are determined, the terms can be combined to form the infinite series.

3. What is the significance of using infinity in a Taylor expansion?

Using infinity in a Taylor expansion allows for a more accurate approximation of the original function. As more terms are added to the infinite series, the approximation becomes closer to the true value of the function.

4. What is the difference between a Taylor expansion in infinity and a regular Taylor expansion?

A regular Taylor expansion involves using a finite number of terms to approximate a function, while a Taylor expansion in infinity uses an infinite number of terms. This allows for a more precise approximation, but also requires more calculation steps.

5. In what fields of science is a Taylor expansion in infinity commonly used?

A Taylor expansion in infinity is commonly used in mathematics, physics, and engineering. It is also used in various other fields such as economics, chemistry, and computer science to approximate functions and make complex calculations easier.

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