Finding the Domain of the Bessel Function Series

[Stratosphere]

Homework Statement

Find domain of $$\sum_{n= 0}^\infty \frac{(-1)^{n}x^{2n}}{2^{2n}(n!)^{2}}$$

Homework Equations

The Attempt at a Solution

I set it all up but I can't really seem to simplify it.

$$\frac{(-1)^{n+1}x^{2(n+1)}}{2^{2n+2}(n+1)!^{2}}\bullet\frac{2^{2n}(n!)^{2}}{(-1)^{n}x^{2n}}$$

 

[Dick]

Why can't you simplify it? What's 2^(2n)/(2^(2n+2))? What's n!/(n+1)!?

 

[Stratosphere]

You mean polynomial division?

 

[Dick]

You mean polynomial division?

No, I mean the law of exponents for the first one and realizing (n)!/(n+1)!=(1*2*3*...*n)/(1*2*3*...*n*(n+1)) for the second one.

 

[Stratosphere]

Explain what the mean by showing me on this much simpler one.

$$\sum_{n= 0}^\infty n!x^{n}$$

Setting it up I get $$\frac{(1+n!)x^{n+1}}{n!x^{n}}$$

Do I cross out the factors?

 

[Dick]

You meant to write (1+n)!, I hope, not (1+n!). Yes, you just cancel the common factors in the numerator and denominator. What do you get?

 

[Stratosphere]

So, I got x+1 but I must have messed something up.

 

[Dick]

So, I got x+1 but I must have messed something up.

Apparently, but not showing how you got that doesn't make it easy to help. What are x^(n+1)/x^n and (n+1)!/n!?