Determining the raidus of convergence

[bigu01]

Homework Statement

Determine the radius of convergence of the given power serie .

Ʃ(x^(2n))/n!

n goes from 0 to infinity

Homework Equations

limit test ratio

The Attempt at a Solution

I am using the limit test ratio and I've got this : [n! * x^(2n+2)]/[(n+1)! * x^2n], then [n!* x^2n * x^2]/[(n+1) * n! * x^2n] , canceling the common things I am left with lim n-> infinity of x^2/n+1, which is 0, but the radius of convergence is infinity, why is infinity?

 

[Dick]

Homework Statement

Determine the radius of convergence of the given power serie .

Ʃ(x^(2n))/n!

n goes from 0 to infinity

Homework Equations

limit test ratio

The Attempt at a Solution

I am using the limit test ratio and I've got this : [n! * x^(2n+2)]/[(n+1)! * x^2n], then [n!* x^2n * x^2]/[(n+1) * n! * x^2n] , canceling the common things I am left with lim n-> infinity of x^2/n+1, which is 0, but the radius of convergence is infinity, why is infinity?

Because if the ratio test gives you a ratio of r then the corresponding radius of convergence is R=1/r. So as r gets very small the radius of convergence gets very large. If you go all the way to r=0 then R=infinity.