Intervals of Convergence- Power Series

[opus]

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?

 

[Orodruin]

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.

 

[opus]

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.

 

[opus]

In rethinking that, I do know that the radius of convergence is R=5. Does this help?

 

[opus]

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.

 

[Orodruin]

You do not know anything about the actual radius of convergence based on what you have posted here.

 

[PeroK]

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?

 

[opus]

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.

 

[PeroK]

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?

 

[opus]

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.

 

[Orodruin]

So now go on arguing about the remaining intervals in similar fashion.

 

[opus]

Ok thanks everyone that was much easier than I originally had thought.