Find the coefficient of $x^{n}$ in the expansion of
$(1 - 2 x + 3 x^{2} - 4 x^{2} + . . .)^{- n}$

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Solution

$S = 1 - 2 x + 3 x^{2} - 4 x^{3} + . . . . . .$

Multiply both sides by $x$

$x S = 0 + x - 2 x^{2} + 3 x^{3} + . . . . . .$

Adding both eqns, we get

$(1 + x) S = 1 - x + x^{2} - x^{3} + . . . . . .$

RHS is infinite GP with common ratio $= - x$

$(1 + x) S = \frac{1}{1 + x}$

$S = (1 + x)^{- 2}$

Now
$(1 - 2 x + 3 x^{2} - 4 x^{3} + . . . . . .)^{- n} = (1 + x)^{2 n}$

Coefficient of $x^{n}$ is

$C_{n}^{2 n} = \frac{(2 n)!}{n! . n!}$

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