Power series

Using the Ratio Test ( where abs = absolute value), we have

L = lim n →∞   abs [((n + 1)³ x²⁽ⁿ⁺¹⁾ ) / 5ⁿ⁺¹] * [ 5ⁿ / (n³ * x²ⁿ) ]   

L = lim n →∞   abs [[ (n+1)³ x² ] / [5 n³]]

L =(1/5) (x² ) *  lim n →∞  [ (n+1)³] /  n³ ] 

Note that  lim n →∞   [ (n+1)³] /  n³ ]  = 1      ......so we have

L = (1/5) (x² )

So if 

(1/5)(x²) < 1     the series converges  ....  and if (1/5)(x²) > 1 the series diverges

Put another way ..... if

(x²) < 5     the series converges  ....  and if (x²) > 5 the series diverges

However....the radius of convergence requires an exponent of 1 on the"x".....so we have

√(x²) < √5   →  abs (x) < √5     and   √(x²) > √5   →  abs (x) > √5 

So the radius of convergence is  R = √5

I believe this is correct......but it has been awhile since I've done this.........if anyone sees any mistakes.....feel free to provide corrections  !!!