Finding Series Radius and Interval of Convergence

In summary, the problem is asking for the series radius and interval of convergence, as well as the values of x for which the series converges absolutely and conditionally. The radius of convergence is determined by taking the limit as x approaches infinity of the ratio of consecutive terms in the series. The series will converge if this limit is less than 1. After finding the radius of convergence, the series can be tested for convergence at the endpoints of the interval. The series will converge absolutely within the radius of convergence, diverge outside of it, and may converge absolutely, conditionally, or diverge at the endpoints. Testing is necessary to determine the behavior at the endpoints.
  • #1
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I am hopelessly confused on a homework assignment.

The problem says " (a) Find the series radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?"

Attached is a sample of problems from the book.

Any help would be appreciated!
 

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  • #2
These are power series. The radius of convergence is

[tex]\lim_{x\to\infty}\left|\frac{a_n_+_1}{a_n}\right|[/tex] then the series converges for


[tex]\lim_{x\to\infty}\left|\frac{a_n_+_1}{a_n}\right|<1[/tex]

After that you find that the series converges say for x in the interval (a,b) and after that try to test whether the series converges at a and b, by letting x=a, and x=b respectively.
 
  • #3
The series converges absolutely inside the radius of convergence, diverges outside and may converge absolutely, converge conditionally, or diverge at the endpoints. That's why you have to test those separately.
 

Related to Finding Series Radius and Interval of Convergence

1. What is the purpose of finding the radius and interval of convergence for a series?

Finding the radius and interval of convergence allows us to determine the range of values for which a given infinite series will converge. This is important because it helps us understand the behavior of the series and whether or not it will have a finite sum.

2. How do you find the radius of convergence for a series?

The radius of convergence, denoted as R, can be found by using the ratio test. This involves calculating the limit of the absolute value of the ratio between the (n+1)th and nth terms of the series. The radius of convergence is then equal to the reciprocal of this limit.

3. What is the interval of convergence?

The interval of convergence is the range of values for which a given series will converge. It is represented by an interval on the real number line, with the center at the series' point of convergence and the radius equal to the series' radius of convergence.

4. Can a series have an infinite radius of convergence?

Yes, it is possible for a series to have an infinite radius of convergence. This means that the series will converge for all values of the independent variable. However, it is more common for series to have a finite radius of convergence.

5. How do you test for convergence at the endpoints of the interval?

To test for convergence at the endpoints of the interval, you can use the endpoint test. This involves plugging in the endpoint values into the original series and checking for convergence or divergence. If the series converges at both endpoints, then the entire interval between the endpoints is included in the interval of convergence.

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