Discovering Maclaurin Series for (1 + x)^(-3) with a Taylor Series Approach

In summary, the best ways to find the Maclaurin series for f(x) = (1 + x)^(-3) are to either make a table and look for a trend in f^(n) or use the expansion of 1/(1+x) and the second derivative of 1/(1+x). However, it is also possible to use the binomial theorem for fractional or negative exponents. Some people suggest making a table as the simpler method.
  • #1
frasifrasi
276
0
I am trying to find the maclaurin series for f(x) = (1 + x)^(-3)

--> what is the best way of doing this--to make a table and look for a trend in f^(n)?
 
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  • #2
...well you can do that but you will just find that is just a binomial series.do you know the nth term in a binomial expansion?
 
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  • #3
Presumably you know the expansion of 1/(1+x) (hint: think geometric series). So what's the second derivative of 1/(1+x), and how does this help? This should give you another way of finding the expansion of (1+x)^(-3).
 
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  • #4
NO! NO! NO! there should be no binomals involved. Is there a simpler way? the way we usually do this is by making a table.

If someone can shed some light, I would appreciate it.
 
  • #5
Well you will have to find f(0),f'(0),f'''(0) and so forth for the traditional method for finding the maclaurin series for that function. But I believe that you should make the table and then make the series if that is the way you know how to do it
 
  • #6
frasifrasi said:
NO! NO! NO! there should be no binomals involved. Is there a simpler way? the way we usually do this is by making a table.

If someone can shed some light, I would appreciate it.
Read my post...
 
  • #7
frasifrasi said:
NO! NO! NO! there should be no binomals involved. Is there a simpler way? the way we usually do this is by making a table.

If someone can shed some light, I would appreciate it.

You can, in fact, extend the binomial theorem to fractional or negative exponents. Morphism suggested an even easier way. You'[ve already been given two very good ways of finding the series. Why don't you appreciate them?
 

1. What is a Taylor series strategy?

A Taylor series strategy is a mathematical approach used to approximate a complex function with a series of simpler functions. It involves expanding a function into an infinite series of its derivatives evaluated at a specific point.

2. How is a Taylor series strategy used in science?

Taylor series strategies are commonly used in many scientific fields, including physics, engineering, and statistics. They are used to approximate complicated functions and analyze their behavior, making them a useful tool in understanding and predicting phenomena.

3. What are the advantages of using a Taylor series strategy?

One advantage of using a Taylor series strategy is its ability to approximate a function with high accuracy, especially when using a larger number of terms in the series. It also allows for a better understanding of the behavior of a function around a specific point.

4. Are there any limitations to using a Taylor series strategy?

While Taylor series strategies can provide accurate approximations, they are limited to functions that are smooth and continuously differentiable. This means they may not work well for functions with sharp discontinuities or singularities.

5. How can I determine the accuracy of a Taylor series approximation?

The accuracy of a Taylor series approximation can be determined by comparing it to the original function. The more terms that are included in the series, the closer the approximation will be to the original function. Additionally, the interval of convergence can also affect the accuracy of the approximation.

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