What is the Interval of Convergence for (x-10)^n/10^n?

In summary, to find the interval of convergence for the series (x-10)^n/10^n, the ratio test is used to obtain (1/10)|x-10| as the radius of convergence. Breaking down the absolute value, we get the interval of convergence as 0 < x < 20. However, when checking the endpoints at x = 0 and x = 20, we need to use a different method since the ratio test is inconclusive when the ratio is 1. The series at x = 0 diverges, while the series at x = 20 converges to 1. Therefore, the final interval of convergence is (0,20).
  • #1
twoski
181
2

Homework Statement



Find the interval of convergence for [itex](x-10)^{n}/10^{n}[/itex]

The Attempt at a Solution



If i use the ratio test on this, i end up with (x-10)/10, which doesn't make sense to me since there is no "n" in this result. Is there another method i need to be using?
 
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  • #2
There shouldn't be an "n"! The ratio test requires that you take the limit as n goes to infinity. The ratio test says that the converges absolutely if that fraction is less than 1. So your condition is that (x- 10)/10< 1.
 
  • #3
twoski said:

Homework Statement



Find the interval of convergence for [itex](x-10)^{n}/10^{n}[/itex]

The Attempt at a Solution



If i use the ratio test on this, i end up with (x-10)/10, which doesn't make sense to me since there is no "n" in this result. Is there another method i need to be using?

Does the series converge/diverge for x = 5? for x=-5? For x = 7? For x = 21? etc., etc.
 
  • #4
twoski said:

Homework Statement



Find the interval of convergence for [itex](x-10)^{n}/10^{n}[/itex]

The Attempt at a Solution



If i use the ratio test on this, i end up with (x-10)/10, which doesn't make sense to me since there is no "n" in this result. Is there another method i need to be using?

After you've applied the absolute ratio test to find your radius of convergence, you get (1/10)|x-10|.

Now we know the radius of convergence is |x-10|<10 breaking down the absolute value we can obtain the interval of convergence :

-10 < x-10 < 10
0 < x < 20

You can check the endpoints at 0 and 20 to see if it converges or not.
 
  • #5
So if i plug in 0, it diverges since it alternates between positive and negative indefinitely.

If i plug in 20, i get 1 since 10^n/10^n is 1. Therefore the test is inconclusive.

So i get (0,20).
 
Last edited:
  • #6
twoski said:
So if i plug in 0, it diverges since it alternates between positive and negative indefinitely.

If i plug in 20, i get 1 since 10^n/10^n is 1. Therefore the test is inconclusive.

So i get (0,20).
I agree with your result, but not with your method. When you check the endpoints, you need to use something other than the ratio test, since you already know that the ratio test is inconclusive when the ratio (its absolute value) is 1.

At the endpoints (i.e., when x = 0 and x = 20) what are the actual series you get? You should have two series that contain only constants - no variables.
 

1. What is an interval of convergence?

An interval of convergence is a range of values for which a given mathematical series will converge. In other words, the series will approach a finite limit as the number of terms in the series approaches infinity within this interval.

2. How do you determine the interval of convergence for a series?

To determine the interval of convergence for a series, you can use a variety of tests such as the ratio test, root test, or alternating series test. These tests involve evaluating the limit of the series and comparing it to known values or criteria to determine if the series converges within a specific interval.

3. Can a series have more than one interval of convergence?

Yes, a series can have multiple intervals of convergence. This occurs when the series converges for different values within a range, but not for the entire range. It is important to note that the endpoints of each interval may or may not be included in the interval of convergence.

4. What happens if a series does not have an interval of convergence?

If a series does not have an interval of convergence, it means that the series does not converge for any values within a given range. This could be due to the series having an infinite number of terms or having no common ratio or pattern among the terms.

5. How is the interval of convergence related to the radius of convergence?

The radius of convergence is the distance from the center of the power series to the closest point where the series converges. The interval of convergence is then the range of values within this distance for which the series converges. In other words, the radius of convergence determines the width of the interval of convergence.

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