Finding the Maclaurin series representation

In summary, to get the function into a form suitable for binomial series, we can rewrite it as ##f(x)=\frac{1}{4}x\left(1-\frac{x}{2}\right)^{-2}## and then expand the second term using the binomial series theorem, multiplying by ##x/4## afterwards.
  • #1
Turion
145
2
Edit: Never mind. Got it.

Homework Statement



[tex]f(x)=\frac { x }{ { (2-x) }^{ 2 } }[/tex]

Homework Equations


The Attempt at a Solution



I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series.

The issue is that binomial series needs to have the form of ##{ (1+x) }^{ k }## but I can't get it into that form. Any idea to get f(x) into that form? The x outside won't go inside the brackets.

Here is the theorem: http://s9.postimg.org/u4qwkrmv3/Binomial_Series.png

Also, my textbook has only one example on binomial series and it is a simpler example.

Attempt:

$$f(x)=\frac { x }{ { (2-x) }^{ 2 } } \\ f(x)=\frac { x }{ 4{ (1-\frac { x }{ 2 } ) }^{ 2 } } \\ f(x)=\frac { 1 }{ 4{ x }^{ -1 }{ (1-\frac { x }{ 2 } ) }^{ 2 } }$$
 
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  • #2
Leave the ##x/4## out in front and expand$$\left (1-\frac x 2\right)^{-2}$$then multiply the ##x/4## back in the result.

[Edit] Apparently you got it while I was typing this.
 

FAQ: Finding the Maclaurin series representation

1. What is a Maclaurin series?

A Maclaurin series is a representation of a function as an infinite sum of terms, where each term is a derivative of the function evaluated at a specific point (usually 0). It is a type of Taylor series that is centered at the origin.

2. Why is it important to find the Maclaurin series representation of a function?

Finding the Maclaurin series representation of a function allows us to approximate the function with a polynomial, making it easier to work with and calculate values. This can be especially useful in situations where the original function is complex and difficult to manipulate.

3. How do you find the Maclaurin series of a function?

To find the Maclaurin series of a function, we can use the formula for a Taylor series and substitute in the specific values for the function and its derivatives at 0. We can also use known Maclaurin series for common functions, such as sin(x) and cos(x).

4. What is the difference between a Maclaurin series and a Taylor series?

A Maclaurin series is a special case of a Taylor series, where the series is centered at the point 0. This means that all the derivatives of the function at 0 are involved in the series, instead of at a general point a as in a Taylor series.

5. Can every function be represented by a Maclaurin series?

Yes, every function that is infinitely differentiable at 0 can be represented by a Maclaurin series. However, the series may not converge for all values of x, so it is important to check for convergence before using the series to approximate the function.

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