Lower bound for radius of convergence of series solutions about a given point

In summary, the problem asks for a lower bound for the radius of convergence of series solutions for a) x_0=0 and b) x_0=2 for the equation (1+x^3)y''+4xy'+y=0. The zero of P(x)=(1+x^3) is -1, so the distance between -1 and 0 is 1, giving a lower bound of 1 for a). For b), the distance between -1 and 2 is 3, but the back of the book gives the answer as sqrt(3). This is because P(x) has three roots, including two complex ones. To find the distance between one of the complex roots and 2, we
  • #1
Esran
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0

Homework Statement



Determine a lower bound for the radius of convergence of series solutions about a) [tex]x_{0}=0[/tex] and b) [tex]x_{0}=2[/tex] for [tex]\left(1+x^{3}\right)y''+4xy'+y=0[/tex].

Homework Equations



N/A

The Attempt at a Solution



The zero of [tex]P\left(x\right)=\left(1+x^{2}\right)[/tex] is [tex]-1[/tex]. The distance between [tex]-1[/tex] and [tex]0[/tex] is [tex]1[/tex], so a) [tex]1[/tex]. The distance between [tex]-1[/tex] and [tex]2[/tex] is [tex]3[/tex], so b) [tex]3[/tex].

The only problem is that for b), the back of the book gives [tex]\sqrt{3}[/tex]. What am I doing wrong?

Thanks in advance for your help!
 
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  • #2
You mean P(x)=(1+x^3). In fact, that has three roots. Two of them are complex, but you have to consider them also. A couple of them are a distance sqrt(3) from x=2.
 
  • #3
Okay, I see now. How do you compute the distance between them and 2 though, in the complex plane?
 
  • #4
One of the roots is cos(pi/3)+i*sin(pi/3), right? Compute the distance from that point to 2+0i.
 

Related to Lower bound for radius of convergence of series solutions about a given point

What is the "lower bound for radius of convergence"?

The lower bound for radius of convergence is the minimum distance from the center of the series expansion where the series will converge. It is also known as the "limiting distance."

Why is the lower bound for radius of convergence important?

The lower bound for radius of convergence is important because it determines the region of convergence for the series expansion. It tells us how close we need to be to the center in order for the series to converge. It also gives us information about the behavior of the series as we move away from the center.

How is the lower bound for radius of convergence calculated?

The lower bound for radius of convergence is typically calculated using the ratio test or the root test. These tests involve taking the limit of the absolute value of the terms in the series and comparing it to a known value. The result will give us the lower bound for the radius of convergence.

Can the lower bound for radius of convergence be negative?

No, the lower bound for radius of convergence cannot be negative. The radius of convergence is a measure of distance, so it cannot be negative. However, it is possible for the radius of convergence to be zero, meaning that the series only converges at the center point.

Does the lower bound for radius of convergence depend on the given point?

Yes, the lower bound for radius of convergence does depend on the given point. The center point of the series expansion will always have a lower bound of zero, but for other points, the lower bound will vary depending on the behavior of the series at that point. It is important to note that the radius of convergence can be different for different points on the same series.

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