Originally Posted by

**HallsofIvy** Post it has already told you that the radius of convergence is 4. That means that the endpoints of the interval of convergence are x= -4 and x= 4.

For x= 4, that series is $\displaystyle \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}4^n$ or $\displaystyle 1+ 2+ \frac{4}{3}+ \frac{24}{5}+ \cdot\cdot\cdot$

That's not going to be easy to decide!

For x= -4 that series is $\displaystyle \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}(-4)^n$. Because that is an 'alternating' series, it will converge as long as the individual terms are eventually decreasing.