Intervals of Convergence- Power Series

In summary, the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, and the interval of convergence is centred on ##4##.
  • #1
opus
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Homework Statement


Hello. I'm not entirely sure what this question is asking me, so I'll post it and let you know my thoughts, and any input is greatly appreciated.

If the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, determine if each of the intervals shown below is a possible interval of convergence for the series.

a) [-4,6]
b) [-6,6)
...etc

Homework Equations

The Attempt at a Solution


Now for power series, (and we have only gone over geometric types) they have a variable so there are three ways that the series can converge.
The series can converge at a single point, over the entire real numbers, or over a given interval (this is the case in which the problem applies).
To determine convergence, we perform a ratio test. However, a ratio test tells us nothing about the convergence or divergence of the series at its endpoints so these must be evaluated individually.

Now for the given question, it says that it does converge at x=6, and gives a possible interval of convergence which is [-4,6). If I understand the question, this means to evaluate the endpoints x=-4 and x=6 separately in the power series.
This is where I'm stuck. So in starting with x=-4, what I want to do is see if ##\sum_{n=0}^\infty a_n((-4)-4)^n## converges and to do so I could just use the divergence test and take the limit of it. But this can't be right because if that was the case, I'd just be taking the limit of ##a_n## multiplied by some constant to the nth power and it would give infinity every time.

Could someone help point me in the right direction?
 
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  • #2
You are overthinking the problem. The question is just whether each given interval is possible or not given the information that x=6 is part of it. You do not need to find the actual interval of convergence.
 
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  • #3
How would I determine that without knowing the actual interval of convergence? Apologies if that is an obvious answer, these have been a little hairy for me.
 
  • #4
In rethinking that, I do know that the radius of convergence is R=5. Does this help?
 
  • #5
I'm not sure if this is an acceptable argument, but with the given series, I believe the center should be at x=4. If we have the interval [-4,6), the radius of convergence would be R=5, and the center would not be at x=4.
 
  • #6
You do not know anything about the actual radius of convergence based on what you have posted here.
 
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  • #7
opus said:
I'm not sure if this is an acceptable argument, but with the given series, I believe the center should be at x=4.

Okay, that's a good start. So, the interval of convergence is centred on ##4##.

What else are you told about the series?
 
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  • #8
The other piece of information is that it converges at x=6. So there is convergence at x=4 and x=6. With this we want to know if the interval [-4,6] is possible.
 
  • #9
opus said:
The other piece of information is that it converges at x=6.

Yes. So, now we know that the interval of convergence is centred on ##4## and contains ##6##.

Does the interval ##[4, 6]## meet those criteria?
 
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  • #10
So in knowing that x=4 is the center and there is convergence at x=6 is a sure thing, [-4,6] can't be an interval because the center of the interval is not x=4.
 
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  • #11
So now go on arguing about the remaining intervals in similar fashion.
 
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  • #12
Ok thanks everyone that was much easier than I originally had thought.
 

Related to Intervals of Convergence- Power Series

What is an interval of convergence for a power series?

An interval of convergence for a power series is the range of values for which the series will converge, or approach a finite value. This means that the terms of the series will become smaller and smaller as the series progresses, eventually reaching a finite value.

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

The interval of convergence for a power series can be determined by using the ratio test or the root test. These tests involve taking the limit of the ratio or root of consecutive terms in the series, and using the resulting value to determine the convergence of the series.

What happens if the series does not converge within the interval of convergence?

If the series does not converge within the interval of convergence, it is considered to be divergent. This means that the terms of the series do not approach a finite value, and the series does not have a sum.

Can the interval of convergence for a power series include its endpoints?

Yes, the interval of convergence for a power series can include its endpoints. This means that the series may converge at one or both of its endpoints, but may diverge at other points within the interval.

How can the interval of convergence for a power series be used in real-world applications?

The interval of convergence for a power series can be used to approximate values of functions. This can be helpful in fields such as physics, engineering, and economics, where precise calculations are necessary. It can also be used to analyze the behavior of functions and make predictions about their values.

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