Introduction to Analysis : Power Series

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Example Questions

Example Question #1 : Power Series

Determine whether the following statement is true or false:

Some power series such as  have a possible radius of convergence that can be found by computing the roots of the coefficients of .

Possible Answers:

False

True

Correct answer:

False

Explanation:

This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.

The proof of one case is as follows:

When  is fixed, , and 

with the assumption that  and  the Root Test is applied to the series .

When  by the assumption and the notation that

implies that  therefore by the Root Test,  does not converge for any  and thus the radius of convergence of  is .

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