A Why Question:Taylor Polynomial of e^x over x?

In summary, the conversation discusses the use of Taylor Polynomials for approximating functions at a specific point, with a focus on the function e^x/x. It is explained that while a Taylor series cannot be used for this function due to a pole at x=0, a Laurent series can be used and is unique. The conversation then touches on the example of \frac{e^x}{x} and clarifies that it is not the same as the Taylor series of e^x. Finally, it is mentioned that this method can be applied to other functions, such as x^2+x/x.
  • #1
RadiantL
32
0
So I was just wondering why when you approximate using the Taylor Polynomials for something like e^x/x at x = 0 you can just find the approximation for e^x and make it all over x, could you do the same for like e^x/x^2 or e^x/x^3?

I hope my question makes sense... thanks
 
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  • #2
You can't find a Taylor polynomial approximation of [itex]\frac{e^x}{x}[/itex] at x=0. The function has a pole at 0. Finding a Taylor approximation at 0, requires the function to exist at that point (and be continuous, differentiable, etc. there).

However, you can make a Laurent series approximation. A Laurent series allows terms like [itex]\frac{1}{x}[/itex],[itex]\frac{1}{x^2}[/itex],... in its expansion.

The Laurent series is unique, so if you found one expression of your function in a Laurent series, then you found it. In our case, we can indeed do

[tex]\frac{e^x}{x}=\frac{1}{x}+1+\frac{x}{2}+...+\frac{x^n}{(n+1)!}+...[/tex]

The same thing will work for [itex]\frac{e^x}{x^2}[/itex] or [itex]\frac{e^x}{x^3}[/itex]. But don't call this a Taylor approximation!
 
  • #3
Oh man, that's funky... my book under integration using the Taylor Polynomials it gives an example

∫e^x/x dx ≈∫ T5(x)/x dx

is that the same thing as the Laurent series?
 
  • #4
Your book is correct. The [itex]T_5(x)[/itex] is indeed the Taylor series of [itex]e^x[/itex] (because [itex]e^x[/itex] exists at x=0 and is smooth there).

However, it would be incorrect to say that [itex]\frac{T_5(x)}{x}[/itex] is the Taylor series of [itex]\frac{e^x}{x}[/itex]. Here you have to use Laurent series.
 
  • #5
Ah, so I'm wrong um so what my book did was just take an approximation for a part of that function? Specifically e^x
 
  • #6
RadiantL said:
Ah, so I'm wrong um so what my book did was just take an approximation for a part of that function? Specifically e^x

Yes, that's exactly what your book did!
 
  • #7
Intresting, so can I do that for everything else? for example...

x^2+x/x could I just take the Taylor polynomial of x^2+x and then make it over x?
 
  • #8
RadiantL said:
Intresting, so can I do that for everything else? for example...

x^2+x/x could I just take the Taylor polynomial of x^2+x and then make it over x?

Yes. But the Taylor polynomial of [itex]x^2+x[/itex] will just be [itex]x^2+x[/itex]... (if you take the degree of the polynomial >1).
 
  • #9
Haha of course, was trying to think up a random example :P Anyway, wow thanks so much your help was very much appreciated. I feel like you should be getting paid for this
 

1. What is a Taylor polynomial?

A Taylor polynomial is a mathematical approximation of a function that is represented as a finite sum of terms. It is used to approximate the value of a function at a given point by using the function's derivatives at that point.

2. What is e^x over x?

e^x over x is an expression that represents the function e^x divided by x. This function is commonly used in mathematics and is known as the exponential integral.

3. Why is the Taylor polynomial of e^x over x important?

The Taylor polynomial of e^x over x is important because it allows us to approximate the value of this function at any point. This can be useful in solving complex mathematical problems or in approximating real-world phenomena.

4. How is the Taylor polynomial of e^x over x calculated?

The Taylor polynomial of e^x over x is calculated using the Taylor series expansion. This involves finding the derivatives of the function at a given point and then plugging them into a formula to calculate the coefficients of the polynomial.

5. What is the significance of using a Taylor polynomial instead of the actual function?

Using a Taylor polynomial instead of the actual function can be beneficial in situations where the function is difficult to evaluate or when we only need an approximation of the function's value. Additionally, the Taylor polynomial can provide a good estimation of the function's behavior within a certain interval.

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