Taylor Series using Geometric Series and Power Series

In summary: Remember that the series is only valid for |x| < 1, so you'll need to include that restriction in your final answer.In summary, the conversation revolved around finding a series for \frac{1}{x^2} and using it to solve a problem. The individual struggled with differentiating and integrating the series, but eventually arrived at the correct solution. The conversation also included a reminder about the validity of the series for a specific range of values.
  • #1
jegues
1,097
3

Homework Statement


See figure attached.


Homework Equations





The Attempt at a Solution



Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the,

[tex]\frac{1}{x^{2}}[/tex]

[tex]\int x^{-2} = \frac{-1}{x} + C[/tex]

How do I deal with the C?

If I can get,

[tex]\frac{-1}{x}[/tex]

I can work with it to get something like the following,

[tex]\frac{\text{first term of geometric series}}{1 - \text{common ratio}}[/tex]

So what do I do about the C? Once I figure this out I can make more of an attempt into shaping,

[tex] \frac{-1}{x}[/tex]

into the form mentioned above.

Any ideas?

Thanks again!
 

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Last edited:
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  • #2
You're going about it backwards. Use

[tex]\frac{1}{x^2} = -\frac{d}{dx}\left(\frac{1}{x}\right)[/tex]
 
  • #3
vela said:
You're going about it backwards. Use

[tex]\frac{1}{x^2} = -\frac{d}{dx}\left(\frac{1}{x}\right)[/tex]

Alrighty I think I've got a series for,

[tex]\frac{1}{x^{2}}[/tex]

See figure attached. Is this correct?

I can't seem to figure out how to express it sigma notation however.

Any ideas?
 

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  • #4
No, it looks like you integrated the series, but you want to differentiate -1/x to get 1/x2.
 
  • #5
vela said:
No, it looks like you integrated the series, but you want to differentiate -1/x to get 1/x2.

Whoops!

How does this look? (See figure attached)
 

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  • #6
Looks good!
 

1. What is a Taylor Series?

A Taylor Series is a mathematical representation of a function as an infinite sum of terms, using the derivatives of the function at a specific point as coefficients. It is used to approximate the value of a function at a given point.

2. What is a Geometric Series?

A Geometric Series is a series of terms where each term is multiplied by a common ratio, known as the common ratio. The sum of a Geometric Series can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio.

3. How are Taylor Series and Geometric Series related?

Taylor Series can be expressed as a Geometric Series if the function being approximated is in the form of a power series. This means that the coefficients of the Taylor Series follow a pattern similar to a Geometric Series, making it easier to calculate.

4. What is a Power Series?

A Power Series is a series of terms where each term is a polynomial with increasing powers of a variable, typically x. It can be used to represent a wide range of functions and can be manipulated to find approximations of those functions.

5. How is the accuracy of a Taylor Series determined?

The accuracy of a Taylor Series depends on the number of terms used in the series. The more terms included, the closer the approximation will be to the actual value of the function. However, using too many terms can lead to computational errors, so it is important to balance accuracy with practicality.

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