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  4. If |x| < 1, then the coefficient of xn in expansion of (1+x+x2+x3+ ldots)2 is:

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

AIEEEAIEEE 2002

Solution:

$\because\left(1+x+x^{2}+x^{3}+\ldots\right)^{2} =\left[(1-x)^{-1}\right]^{2}$
$=(1-x)^{-2}$
Coefficient of $x^{n}$ in $\left(1+x+x^{2}+\ldots\right)^{2}$
= coefficient of $ x^{n} \text { in }(1-x)^{-2} $
$=^{n+2-1} C_{2-1}={ }^{n+1} C_{1} $
$=n+1$