How Do You Determine the Interval of Convergence for a Series?

In summary, the given series has a radius of convergence of 4 and converges for -4 < x < 4. The end points should be checked to determine if they are included in the interval of convergence.
  • #1
knv
17
0
1. Find the radius and the interval of convergence for the series:

Ʃ n=2 --> inf : [(-1)nxn]/ [4nln(n)]





2.To find the radius, we use the alternating series test. **an+1/an




3. From the alternating series test I find that the limit as n --> inf = 4. So our radius is 4. Although I do not know how to get the intervals from the radius. Can anyone help me?

Would we just plug in ±4 for x and solve for convergence or divergence?
 
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  • #2
knv said:
1. Find the radius and the interval of convergence for the series:

Ʃ n=2 --> inf : [(-1)nxn]/ [4nln(n)]





2.To find the radius, we use the alternating series test. **an+1/an
You mean the ratio test.
3. From the alternating series test I find that the limit as n --> inf = 4. So our radius is 4. Although I do not know how to get the intervals from the radius. Can anyone help me?

Would we just plug in ±4 for x and solve for convergence or divergence?

Yes. You know you have convergence for ##-4 < x < 4##. So just check to see which, if any, of the end points to include.
 

FAQ: How Do You Determine the Interval of Convergence for a Series?

What is the Interval of Convergence?

The Interval of Convergence is the range of values for which a given power series will converge. It is typically denoted by [a,b] and includes all values of x within that range for which the series will converge.

How is the Interval of Convergence determined?

The Interval of Convergence can be determined by applying the Ratio Test or the Root Test to the given power series. These tests involve taking the limit of the ratio or root of consecutive terms in the series and evaluating the result to determine if the series converges or diverges.

What happens if a value falls outside the Interval of Convergence?

If a value falls outside the Interval of Convergence, the series will either diverge or have an indeterminate result. This means that the series does not converge at that particular value and cannot be used to calculate the sum of the series.

Can the Interval of Convergence change?

Yes, the Interval of Convergence can change depending on the given power series. It may be different for different series, even if they have the same general form. It is important to calculate the Interval of Convergence for each series individually.

How is the Interval of Convergence used in real-world applications?

The Interval of Convergence is used in a variety of scientific and mathematical fields, including physics, engineering, and finance. It is used to determine the accuracy and reliability of numerical methods and models, and to analyze and solve complex equations and problems.

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