`sum_(n=1)^oo (4x)^n/n^2`

To find radius of convergence of a series `sum` `a_n` , apply the Ratio Test.

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

`L=lim_(n->oo) | (4x)^(n+1)/(n+1)^2 * n^2/(4x)^n|`

`L= lim_(n->oo) |(4xn^2)/(n+1)^2|`

`L = |4x| lim_(n->oo) |n^2/(n+1)^2|`

`L = |4x| * 1`

`L = |4x|`

`L =4|x|`

Take note that in Ratio Test, the series converges when L < 1.

`L < 1`

`4|x| lt 1`

`|x|lt1/4`

**Therefore, the radius of convergence of the given series is `R = 1/4` .**