(D) This is an $A.G.P.$ (Arithmetic-Geometric Progression).Let $S = 1 + 2x + 3x^2 + 4x^3 + \dots \infty$ $(1)$Multiply by $x$:$xS = x + 2x^2 + 3x^3 +

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(D) This is an $A.G.P.$ (Arithmetic-Geometric Progression).Let $S = 1 + 2x + 3x^2 + 4x^3 + \dots \infty$ $(1)$Multiply by $x$:$xS = x + 2x^2 + 3x^3 + \dots \infty$ $(2)$Subtracting $(2)$ from $(1)$:$S - xS = 1 + (2x - x) + (3x^2 - 2x^2) + (4x^3 - 3x^3) + \dots \infty$$(1 - x)S = 1 + x + x^2 + x^3 + \dots \infty$Since $|x| $(1 - x)S = \frac{1}{1 - x}$$S = \frac{1}{(1 - x)^2}$