Finding Radius of Convergence: Ratio Test for Series

[Fiz2007]

Homework Statement

find the interval of convergence of
\sum[(2k+1)!/((2^k)((k!)²)]* [x^k]

Homework Equations

Ratio Test

The Attempt at a Solution

I already found that it converges on (-1/2, 1/2) by using power series with b=0 and testing the rest of it as a_k. However, I am unsure about the end points. I tried using the ratio test with x = (-1/2) and I get the limit to be 1 so the test fails. What test would I be able to use? Any help is greatly appreciated.

 

[Fiz2007]

And i have no clue why the summation is putting all the ^{stuff in there but it is suppose to say just what is in the text at the end.}

 

[Dick]

A series can't converge if the limit of the terms doesn't go to zero. Try that test at the endpoints.