Question

Sum to infinite terms of
$1+3x+5x^2+7x^3+\dots \dots +\infty (Hf|x|<1)$ is

Answer 1

Let $S=1+3x+5x^2+7x^3+...$

$xS=x+3x^2+5x^3+...$

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$S(1-x)=1+2x+2x^2+2x^3+...$ [subtracting]

$1+s(1-x)=2(1+x+x^2+...)=\frac{2}{1-x}$

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

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