The coefficient of x^ 2012 in 1+x (1+x^2 )(1-x) is - Vidyadip

Question

The coefficient of $x^{2012}$ in $\frac{1+x}{\left(1+x^2\right)(1-x)}$ is

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Answer

a
(a)

We have, $\frac{1+x}{\left(1+x^2\right)} \frac{(1-x)}{}$

$\frac{1+x}{\left(1+x^2\right)(1-x)}=\frac{x}{1+x^2}+\frac{1}{2\left(1+x^2\right)}$

$+\frac{1}{2(1-x)}$

$=x\left(1+x^2\right)^{-1}+\frac{1}{2}\left(1+x^2\right)^{-1}+\frac{1}{2}(1-x)^{-1}$

Coefficient of $x^{2012}$ in

$\left[x\left(1+x^2\right)^{-1}+\frac{1}{2}\left(1+x^2\right)^{-1}+\frac{1}{2}(1-x)^{-1}\right]$ is

$=0+\frac{1}{2}+\frac{1}{2}=1$

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