For the power series `sum_(n=0)^oo (2n)!(x/3)^n` , we may apply Ratio Test.

In **Ratio test**, we determine the limit as:

`lim_(n-gtoo)|a_(n+1)/a_n| = L`

or

`lim_(n-gtoo)|a_(n+1)*1/a_n| = L`

Then ,we follow the conditions:

a) `L lt1` then the series **converges absolutely.**

b) `Lgt1 ` then the series **diverges**.

c) `L=1` or does not exist then the **test is
inconclusive**.The series may be divergent, conditionally convergent, or
absolutely convergent.

The given power series `sum_(n=0)^oo (2n)!(x/3)^n` has:

`a_n =(2n)!(x/3)^n`

Then,

`1/a_n=1/((2n)!)(3/x)^n`

` =1/((2n)!)(3^n/x^n)`

` =3^n/((2n)!x^n)`

`a_(n+1) =(2(n+1))!(x/3)^(n+1)`

` = (2n+2)!x^(n+1)/3^(n+1)`

`= (2n+2)(2n+1)((2n)!) x^n*x/(3^n*3)`

`=((2n+2)(2n+1)((2n)!) * x^n*x)/(3^n*3)`

Applying the Ratio test on the power series, we set-up the limit as:

`lim_(n-gtoo) |((2n+2)(2n+1)((2n)!) * x^n*x)/(3^n*3)3^n/((2n)!x^n)|`

Cancel out common factors: `x^n` , `(2n)!` , and `3^n` .

`lim_(n-gtoo) |((2n+2)(2n+1)*x)/3|`

Evaluate the limit.

`lim_(n-gtoo) |((2n+2)(2n+1)*x)/3| = |x/3|lim_(n-gtoo) |(2n+2)(2n+1)|`

`= |x/3|* oo`

` = oo `

The limit value `L= oo` satisfies` Lgt 1` for all `x` .

Therefore, the power series `sum_(n=0)^oo (2n)!(x/3)^n`
**diverges for all x** based from the Ratio test
criteria: `Lgt1 ` then the series diverges.

There is **no interval for convergence**.

Note: The **radius of convergence** is `0` . The `x=0`
satisfy the convergence at points where `(2n)!(x/3)^n=0` .

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