Power series and integration >

In summary, the conversation is about the topic of power series and integration, specifically regarding a question and solution involving a function and a power series. The person asks if it is always possible to manipulate a function and a power series in such a way, and the other person explains that it is not technically a power series since it includes negative powers.
  • #1
FaNgS
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Power series and integration >:(

Hello, this question is about power series and integration of power series, the question and my working is on the image below.
I had to write the question and my working plus the correct answer on a piece of paper and scan it, sorry for the hasle

http://i148.photobucket.com/albums/s38/InsertMe/img015.jpg

Smilies in the image :P
 
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  • #2
[tex]-\frac{1}{x} - \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n-1}}{4n^2-1}[/tex]

[tex]= -\frac{1}{x} + \sum_{n=0}^{\infty}\frac{-(-1)^nx^{2n-1}}{4n^2-1}[/tex]

[tex]= -\frac{1}{x} + \sum_{n=0}^{\infty}\frac{(-1)^{n+1}x^{2n-1}}{4n^2-1}[/tex]

[tex]= -\frac{1}{x} + \frac{(-1)^{0+1}x^{2\cdot 0-1}}{4\cdot 0^2-1} + \sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{2n-1}}{4n^2-1}[/tex]

[tex]= -\frac{1}{x} + \frac{x^{-1}}{1} + \sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{2n-1}}{4n^2-1}[/tex]

[tex]=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{2n-1}}{4n^2-1}[/tex]
 
  • #3
thanks a lot AKG

just one question is it always possible to do that whenever i get a function plus (or minus) a power series?
 
  • #4
In this particular case you should have noticed that when n= 0, x2n-1 is just x-1, the same as the 1/x you are adding. Strictly speaking, what you have is not a "power series" since the standard definition requires non-negative powers only.
 

1. What is a power series?

A power series is an infinite series of the form ∑an(x-a)n, where a is a constant and x is the variable. The coefficients an represent the varying powers of x, and the series converges to a function f(x) when the values of x fall within a certain interval.

2. How is a power series used in integration?

Power series can be used in integration by representing a function as a power series and then integrating each term. This can be helpful in solving certain types of integrals, such as those involving trigonometric functions or logarithms.

3. What is the relationship between a power series and its radius of convergence?

The radius of convergence of a power series is the distance from the center point (a) to the nearest point where the series converges. The larger the radius of convergence, the more terms of the series are needed to approximate the function accurately.

4. Can power series be used to approximate any function?

No, power series can only approximate functions that are continuous and differentiable within the interval of convergence. The accuracy of the approximation also depends on the radius of convergence.

5. How do you determine the radius of convergence of a power series?

The radius of convergence can be determined by using the ratio test or the root test. These tests involve taking the limit of the ratio or root of the absolute values of the terms in the series. If the limit is less than 1, the series converges, and the radius of convergence is the distance from the center point to the point where the limit is taken. If the limit is greater than 1, the series diverges.

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