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\text { If }|x|<1 \text {, } then the coefficient of xn in the expansion of \left(1+2 x+3 x^2+4 x^3+\ldots\right)^{1 / 2} \text { is }

Option: 1

1


Option: 2

2


Option: 3

n


Option: 4

n+1


Answers (1)

best_answer

\text { : Since } 1+2 x+3 x^2+4 x^3+\ldots \infty=(1-x)^{-2} \text {, we have }\begin{aligned} & \left(1+2 x+3 x^2+4 x^3+\ldots \infty\right)^{1 / 2}=\left[(1-x)^{-2}\right]^{1 / 2} \\ & =(1-x)^{-1} \\ & =1+x+x^2+\ldots+x^n+\ldots \infty \end{aligned}Therefore the coefficient of xn is 1

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Devendra Khairwa

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