Find the Maclaurin series for the function using definition of a Mclaurin series

Find the Maclaurin series for the function using definition of a Mclaurin series. Also, find the radius of convergence.

Re: Find the Maclaurin series for the function using definition of a Mclaurin series

This should be the Mclaurin series?

How do I find the radius of convergence?

Re: Find the Maclaurin series for the function using definition of a Mclaurin series

For power series the simplest way to find the radius of convergence is to use the "ratio test".

A series of positive numbers, will converge as long as the sequence has a limit less than 1. (If that limit is larger than 1, it diverges. If it is equal to 1, we don't know.)

Here, . Those terms may not be positive but we know that a power series always converges **absolutely** inside its radius of convergence so we can take the absolute value:

What is the limit of that as n goes to infinity? For what values of x is that limit less than 1?

Re: Find the Maclaurin series for the function using definition of a Mclaurin series

Not sure I'm doing this right?

Re: Find the Maclaurin series for the function using definition of a Mclaurin series

Re: Find the Maclaurin series for the function using definition of a Mclaurin series

Since the ratio test = 0 then we don't know?

Re: Find the Maclaurin series for the function using definition of a Mclaurin series

Re: Find the Maclaurin series for the function using definition of a Mclaurin series