It is known :
x + x² + x³ + x⁴ + .... = x/1 - x
Differentiating both sides :
1 + 2x + 3x² + 4x³ + .... = 1/(1-x)^{2   
Miultiplying both sides by x :
x + 2x2 + 3x3 + 4x4 + .... = x/(1-x)2
Differentiating again :
1 + 4x + 9x2 + 16x4 + .... = (x+1)/(1-x)2
Multiplying by x :
1 + 4x2 + 9x3 + 16x5 + .... = x(x+1)/(1-x)3}
 

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