How to Find the Power Series and Radius of Convergence for Arctan'x

In summary, the conversation discusses how to expand arctan'x into a power series and how to find the power series for it. The formula for arctan'x is 1/(1+x^2) and the series can be written as 1 - x^2 + x^4 - x^6 +...+ (-1)^n x^{2n}. The participants also discuss how to find the radius of convergence, which is determined to be |x|<1 using methods such as integrating the series, the ratio test, and the root test. It is also noted that the expression 1/(1+x^2) is defined for all complex x except i or -i, and the radius of convergence
  • #1
mathusers
47
0
how could i expand something such as arctan'x into a power series. also how would you be able to find the power series for it?so far i have managed to work out that:

arctan'x = [itex] \frac{1}{1 + x^2} [/itex]

[itex]\frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n}[/itex]

how do you work out the radius of convergence though: i know it is : |x|< 1.. but how do you work it out please?
 
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  • #2
Integrate the series you wrote term by term. Watch out for the first line you wrote. You're missing a derivative operator acting on the "arctan" function.
 
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  • #3
just to clear confusion.. i did mean the derivate of arctanx .. i.e. d/dx arctan x , hence arctan'x...

how would i show the radius of convergence as |x|<1 though please?

to work it out i tried it on
[itex](-1)^n x^{2n}[/itex]
i ended up with

[itex]a_{n+1} / a_{n} = \frac{|x|^{2n + 2}}{|x|^{2n}} = |x|^2/1 [/itex] as n tends to infinity... ...

so radius of convergence is |x|< 1...

is this working out correct?
 
  • #5
the way my book (Stewart) does it is they say that since it's a geometric series the series will be convergent when |-x^2n|<1 = x^2<1=|x|<1
 
  • #6
That's the ratio test at work. The alternating series test also works here.
 
  • #7
Another way to check would have been to see where the expression [itex]\frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n}[/itex] is valid, since that is the basis for the new power series. We can see that the expression fails for values of x larger than 1. Really, its just a tiny variation of what DH and dex said :(
 
  • #8
In general, a power series will converge as long as has no reason not too!

[tex]\frac{1}{1+x^2}[/tex] is defined for all complex x except i or -i. The radius of the "disk" of convergence in the complex numbers is 1 so, restricting to the real numbers, the radius of the interval of convergence is also 1.

Of course, you can look at it as a geometric series: it is of the form arn with a= 1, r= -x2: its sum is [tex]\frac{1}{1+x^2}[/tex] and it converges as long as |-x2|< 1 or |x|< 1.

Similarly, the ratio test gives the same result: |x|< 1.

Oh, and the root test: [itex]^n\sqrt{a_n}= |x|< 1[/itex] as well.

I think we have determined that the radius of convergence is 1!
 

1. What is a power series of arctan'x?

A power series of arctan'x is a mathematical representation of the arctangent function, which is the inverse of the tangent function. It is an infinite series of terms involving powers of x, and it is used to approximate the value of arctan'x for any given value of x.

2. How is a power series of arctan'x derived?

A power series of arctan'x is derived by expanding the arctangent function into an infinite series using the Taylor series expansion. This involves finding the derivatives of the function at a specific point and plugging them into the formula for a Taylor series.

3. What is the radius of convergence for a power series of arctan'x?

The radius of convergence for a power series of arctan'x is 1, which means that the series will converge within a distance of 1 unit from the center point. This can be seen from the formula for the radius of convergence, which is given by R = lim |an|1/n.

4. What is the significance of using a power series to approximate arctan'x?

The significance of using a power series to approximate arctan'x is that it allows us to calculate the value of the arctangent function for any given value of x. This is especially useful when dealing with complex mathematical problems where an exact solution is not easily attainable.

5. How accurate is a power series approximation of arctan'x?

The accuracy of a power series approximation of arctan'x depends on the number of terms used in the series. The more terms included, the more accurate the approximation will be. However, since it is an infinite series, it will always be an approximation and not an exact value.

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