All Introduction to Analysis Resources
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 .
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 .