Find a power series representation for the function and determine the interval of convergence
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What we need to do here is to relate this function back to (2). This is actually easier than it might look. Recall that the x in (2) is simply a variable and can represent anything. So, a quick rewrite of gives (x),
g(x) = 1/1 - (x raise to power 3)
and so the -x raise to power 3 in holds the same place as the x in (2). Therefore, all we need to do is replace the x in (3) and we’ve got a power series representation for g(x).
What we need to do here is to relate this function back to (2). This is actually easier than it might look. Recall that the x in (2) is simply a variable and can represent anything. So, a quick rewrite of gives (x),
g(x) = 1/1 - (x raise to power 3)
and so the -x raise to power 3 in holds the same place as the x in (2). Therefore, all we need to do is replace the x in (3) and we’ve got a power series representation for g(x).
Notice that we replaced both the x in the power series and in the interval of convergence.
All we need to do now is a little simplification.
So, in this case, the interval of convergence is the same as the original power series. This usually won’t happen. More often than not the new interval of convergence will be different from the original interval of convergence.
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