Sun,

your problem is pretty straightforward you have a power series. Σ_{n=0}^{∞} nx^n.

First thing you need to find out if the series is convergent and, if so, for what value of x. I think you should be able to prove that it converges for -1<x<1. Note that for x=1 you would get the series Σ_{n=0}^{∞} n which does not converge.
Show what happens for x=-1.... After that, when x is inside the range (-1, 1) .. the derivative of the sum is equal to the sum of the derivatives of its terms. Now, the first derivative of the generic term of the series, nx^{n}, is n*nx^{n-1}
=n^{2}x^{n-1}. So with simple algebra you get your answer for y', i.e.,

y' = Σ_{n=1}^{∞} n^{2}x^{n-1 }=1x^{0} +2^{2}x^{1}+3^{2}x^{2}+....+(n+1)^{2}x^{n}+ ..

Repeat the same argument starting from y' this time and its defining series to compute y''. Do not forget to discuss the convergence issues or you could not be doing the derivation term by term.