Representing a function as a power series

In summary, the conversation discusses evaluating the indefinite integral as a power series and finding the radius of convergence for the given function. One approach is to expand arctan(x) as a power series around 0 and then integrate, with the necessary factor of (-1)^n included.
  • #1
grothem
23
1

Homework Statement


Evaluate the indefinite integral as a power series and find the radius of convergence

[tex]\int\frac{x-arctan(x)}{x^3}[/tex]


I have no idea where to start here. Should I just integrate it first?
 
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  • #2
Sure, you could do that. But, I think what they want to do is expand arctan(x) as a power series around 0 and then integrate.
 
  • #3
ok. So arctan(x) = [tex]\int\frac{1}{1+x^2}[/tex]
= [tex]\int\sum (x^(2*n))[/tex]
= [tex]\sum\frac{x^(2(n+1)}{2(n+1)}[/tex]

is this what you mean?
 
  • #4
That's one way to get a series for arctan, yes. But you forgot a (-1)^n factor. The expansion of 1/(1-x) has all plus signs. 1/(1+x) doesn't.
 

1. What is a power series?

A power series is a type of infinite series in mathematics that represents a function as a sum of terms, where each term is a constant multiplied by a variable raised to a power.

2. How is a power series used to represent a function?

A power series is used to approximate a function by breaking it down into simpler polynomial functions. By adding more and more terms in the power series, the approximation becomes more accurate.

3. What is a Maclaurin series?

A Maclaurin series is a special type of power series where the center of the series is at x=0. It is named after Scottish mathematician Colin Maclaurin.

4. How do you find the coefficients in a power series?

The coefficients in a power series can be found by using the Taylor series formula, which involves taking derivatives of the function at the center of the series. Alternatively, the coefficients can also be determined using a recurrence relation.

5. What are the applications of representing a function as a power series?

Power series are commonly used in calculus, physics, and engineering to approximate complicated functions and solve differential equations. They are also used in computer algorithms to calculate values of mathematical functions.

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