Radius / Interval of Convergence (Power Series)

In summary, the conversation is about finding the radius and interval of convergence for the power series given by the sum from n=0 to infinity of (2-n^(1/2))*(x-1)^(3n). The person is unsure about how to use the ratio test and is questioning whether 1/|x-1|^3 < 1 is the correct result. After clarifying that only "n" is raised to the power of 1/2, the conversation shifts to discussing the interval and radius of convergence. The person is still unsure about the ratio test and is seeking help. The summary concludes with the person thanking for the assistance.
  • #1
student45
I need help finding the radius & interval of convergence of the following power series:

The sum from n=0 to infinity of...

(2-(n)^(1/2)) * (x-1)^(3n)

I think the ratio test is supposed to work, but I narrow the limit of the test down to 1/|x-1|^3 < 1, and this doesn't make sense to me. Is there an easier way, or am I doing something wrong?

Thanks.
 
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  • #2
[tex] \sum_{n=0}^{\infty} (2-n^{\frac{1}{2}})(x-1)^{3n} [/tex]

Is this the question?
 
Last edited:
  • #3
Yeah, that's the one. Is the ratio test the right approach? Is 1/|x-1|^3 < 1 the right result? I'm just not sure where to go from there, or even if that is right for the limit. Thanks a lot for your help.
 
  • #4
Oh except that only the "n" is raised to 1/2. Not (2-n). The quantity is (2-n^.5)
 
  • #5
[tex] \sum_{n=0}^{\infty} (2-n^{\frac{1}{2}})(x-1)^{3n} [/tex]

Now that's right. Interval/Radius of convergence is the issue. I just don't know what I'm doing wrong with the ratio test.

Thanks.
 
  • #6
Using the ratio test:

[tex] (x-1)^{3}\; \frac{2-(n+1)^{\frac{1}{2}}}{2-n^{\frac{1}{2}}} [/tex] As [tex] n\rightarrow \infty [/tex] we get [tex] |(x-1)^{3}| < 1 [/tex]
 
  • #7
I appreciate it. Thanks!
 

What is the radius of convergence of a power series?

The radius of convergence of a power series is the distance from the center of the series to the point where the series converges. It is represented by the letter "R" and is typically a positive real number or infinity.

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

To determine the radius of convergence of a power series, you can use the ratio test or the root test. These tests involve taking the limit of the ratio or root of consecutive terms in the series. If the limit is less than 1, the series converges, and the radius of convergence is the absolute value of the value of the variable at which the series converges. If the limit is greater than 1, the series diverges, and the radius of convergence is 0. If the limit is equal to 1, further tests are needed to determine the convergence or divergence of the series.

What is the interval of convergence of a power series?

The interval of convergence of a power series is the set of all values of the variable for which the series converges. It is important to note that the endpoints of the interval may or may not be included in the interval of convergence.

How do you find the interval of convergence of a power series?

To find the interval of convergence of a power series, you can use the ratio test or the root test to determine the radius of convergence. Then, you can use the center and radius to determine the endpoints of the interval. Finally, you can test the convergence of the series at these endpoints to determine if they should be included in the interval of convergence or not.

What happens when the radius of convergence of a power series is 0 or infinity?

If the radius of convergence of a power series is 0, the series will converge only at the center of the series. If the radius of convergence is infinity, the series will converge for all values of the variable. In both cases, the interval of convergence is a single point.

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