What Is the Interval of Convergence for the Given Series?

[Ki-nana18]

Homework Statement

The summation from n=1 to infinity of ((n!)x^(2n))/((2n-1)!) Find the Interval of Convergence of this series.

Homework Equations

Ratio test

The Attempt at a Solution

I applied the ratio test, then got x^2 times the limit as n approaches infinity of (n+1)/(2n(2n+1)). I took the limit and got zero and since 0<1 the series converges. Does this mean the interval of convergence is (-infinity,+infinity)?

 

[Dick]

Yes, the limit of the ratio is zero. So that's exactly what it means. The interval of convergence is (-infinity,infinity).