1.) Find the open interval of convergence of the series:
SOLUTION 1.)
In order to find the open interval of convergence of the series, use the Ratio Test to determine the radius of convergence. Recall the Ratio Test:
So, let's calculate our ratio:
Cancelling appropriate terms leaves:
Therefore, the radius of convergence 2. Thus the series converges for the interval:
In order to be entirely correct on our interval of convergence, we would need to test for whether or not our series converges at the endpoints. In order to do so, we would plug our endpoints into our original power series and create two new infinite series to analyze. However, the ratio test will not work for our new series. So, we need to learn about other tests of convergence before we can determine the behavior of the endpoints.
2.) Find the open interval of convergence of the series:
SOLUTION 2.) In order to find the open interval of convergence of the series, use the Ratio Test to determine the radius of convergence. Let's calculate the Ratio Test:
Cancelling appropriate terms leaves:
Therefore, since for any x, the limit goes to zero (which is less than 1)
converges for all x. Its radius of convergence is 
.
3.) Find the open interval of convergence of the series:
SOLUTION 3.) Once again, let's calculate the Ratio Test:
Next, simplify terms:
Cancelling appropriate terms leaves:
So, we get:
Therefore, our open interval of convergence for 
is
with a radius of 
.