Find the generating functions of the series $1 - 3 x + 5 x^{2} - 7 x^{3} + 9 x^{4} - 11 x^{5} + . . . .$

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Solution

$S = 1 - 3 x + 5 x^{2} - 7 x^{3} + 9 x^{4} . . . . . . . x S = x - 3 x^{2} + 5 x^{3} - 7 x^{4} + 9 x^{5} . . . . . . S (1 + x) = 1 - 2 x + 2 x^{2} - 2 x^{3} + 2 x^{4} - 2 x^{5} . . . . . . . .$
Upto infinity
$S (1 + x) = 1 + 2 (- 2 x) + 2 {(- 2 x)}^{2} + 2 {(- x)}^{3} + 2 {(- x)}^{4}$
Upto infinity
$= 1 + 2 ((- x) + {(- x)}^{2} + {(- x)}^{3})$
Upto infinity
$= 1 + 2 (a + a r + a r^{2} + . . . . . . .) = 1 + 2 (\frac{a}{1 - r}) = 1 + 2 (\frac{- x}{1 + x}) S (H x) = \frac{1 - x}{1 + x} S = \frac{1 - x}{{(1 + x)}^{2}}$

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