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If | x |< 1 , then the coefficient of $x^{n}$ in expansion of ${(1 + x + x^{2} + x^{3} + . . . . . .)}^{2}$ is

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n-1

n+2

n+1

answer is D.

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Detailed Solution

$\begin{array}{rcl} = & {(1 + x + x^{2} + x^{3} + . . . . .)}^{2} \\ = & {(1 - x)}^{- 2} = 1 + 2 x + 3 x^{2} + \cdot \cdot \cdot (n + 1) x^{n} + 1 \cdot \cdot \cdot \end{array}$

Coeff $x^{n} = n + 1$

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