We can apply the ratio test, by evaluating:
abs (a_(n+1)/a_n) = abs( ( ( (2(n+1))!x^(n+1))/((n+1)^2!)) / ( ( (2n)!x^n)/((n^2)!)) )∣∣∣an+1an∣∣∣=∣∣
∣
∣∣(2(n+1))!xn+1(n+1)2!(2n)!xn(n2)!∣∣
∣
∣∣
abs (a_(n+1)/a_n) = abs( x^(n+1)/x^n) ( (2(n+1))!) / ((2n)!) (n^2!)/((n+1)^2!) ∣∣∣an+1an∣∣∣=∣∣∣xn+1xn∣∣∣(2(n+1))!(2n)!n2!(n+1)2!
abs (a_(n+1)/a_n) = abs( x ) ((2n+2)!) / ((2n)!) (n^2!)/((n^2+2n+1)!) ∣∣∣an+1an∣∣∣=|x|(2n+2)!(2n)!n2!(n2+2n+1)!
abs (a_(n+1)/a_n) = abs( x ) ((2n+2)(2n+1))/ ((n^2+2n+1)(n^2+2n)* ... * (n^2+1))
Clearly:
lim_(n->oo) abs (a_(n+1)/a_n) = abs(x)*lim_(n->oo) ((2n+2)(2n+1))/ ((n^2+2n+1)(n^2+2n)* ... * (n^2+1)) = 0
for every x, thus the series is convergent for x in RR.