Evaluating Indefinite Integral as Power Series: Find Radius of Convergence

In summary, the conversation is about evaluating the indefinite integral (x-tan^-1x)/x^3 dx as a power series and finding the radius of convergence. The speaker has found the power series of tan^-1x = x^(2n+1) (-1)^n/(2n+1) and is unsure of how to continue with the integration. They are seeking advice and simpler explanations on the topic. The suggested steps are to pull the integral apart into two integrals, integrate the first one, write the series expansion of arctan and divide its terms by x^3, integrate by parts, and think of a way to find the convergence radius. It is mentioned that the integral can also be solved exactly by
  • #1
jaidon
42
0
Evaluate the indefinite integral as a power series and find radius of convergence. (i don't know how to type the integral and summations signs, sorry)


(integral sign) (x-tan^-1x)/x^3 dx. ( if you write this out it makes more sense)

i was able to find the power series of tan^-1x = x^(2n+1) (-1)^n/(2n+1).
i don't know how to continue on with this. all we have learned is to use the power series of the geometric series 1/(1-x), and some integration/differentiation methods.

i am rather confused on the whole topic, so if anyone has any ideas, the simplest explanations would be greatly appreciated. thanks
 
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  • #2
So your integral is
[tex] \int \frac{x-\arctan x}{x^{3}} dx [/tex]

??Okay,for term by term integration of it,separate it into 2 integrals...Though it's not really helpful for the convergence part...

Daniel.
 
  • #3
that is the integral, thanks, but i am not sure what to do after pulling it apart into two integrals. quite honestly, i am puzzled by this whole topic. any advice?
 
  • #4
1.Pull apart into integrals.
2.Integrate the first.It's elementary.
3.Write the series expansion of "artan" and divide its terms by x^{3}.
4.Integrate by parts eery term of the new series...
5.Think of a way to get the convergence radius.

Daniel.

P.S.It can be done exactly (find the antiderivative).
 

1. What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the antiderivative of a function. It is the inverse operation of taking a derivative, and it is used to find the original function when only its derivative is known.

2. What is a power series?

A power series is an infinite series that represents a function as a sum of powers of the variable. It is written in the form of f(x) = a0 + a1x + a2x2 + a3x3 + ..., where an is a coefficient and x is the variable.

3. How is an indefinite integral evaluated as a power series?

To evaluate an indefinite integral as a power series, we use a process called integration by substitution. This involves substituting the variable in the integral with a new variable, and then using the power series representation of the function to solve the integral.

4. What is the radius of convergence?

The radius of convergence is a measure of how far away from the center of a power series a function can be evaluated and still have a convergent series. It is represented by the variable R and is determined by the coefficients of the power series.

5. Why is it important to find the radius of convergence?

Finding the radius of convergence is important because it tells us the range of values for which the power series is valid and can be used to approximate the original function. If a value is outside the radius of convergence, the power series will not converge and cannot be used to evaluate the function.

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