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Radius of convergence

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is.

a) centered at and

NOTE: I know the function has a removable singularity at 0 and it can be redefined to the following:

But I'm confused by how to represent the Taylor series. I do know the power series representation for this and I believe that it converges between (-1,1) with radius 1. I'm not sure if this applies or not.

b) ln(x), centered at and

c) centered at

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The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is.

a) centered at and

NOTE: I know the function has a removable singularity at 0 and it can be redefined to the following:

Solution. If we consider the function , then
(1) We can easily get its Taylor series . Since ...

Solution Summary

This shows how to determine the radius of convergence of the Taylor Series for functions.

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