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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?