Tasty Series

Consider the series $y=x+x^2+x^3+x^4+.......$ differentiating w.r.t. x $\frac{dy}{dx}=1+2x+3x^2+4x^3+....$ multiplying x then again differentiating $\frac{d(x\frac{dy}{dx})}{dx}=1+2^2x+3^2x^2+4^2x^3+..........$ multiplying x and placing x=2 we get the required series as $x\frac{d(x\frac{dy}{dx})}{dx}=1.1+2^2.2^2+3^2.2^3+4^2.2^4+......$ putting $y=x\frac{(1-x^n)}{1-x}$ we get $sum=2^{n+1}(n).(n-2)+6.(2^n-1)$ put n=9 to get the desired result as $\boxed{67578}$

It is just : 1 2 + 4 2^2 + 9 * 2^3+ ... + 9^2 + 2^9 = 67578