<s>[INST] How to show that $ 1 + x + x^2 +x^3 +...= \frac{1}{1-x} $? By long division, it is easy to show that $$ \frac{1}{1-x} = 1 + x + x^2 +x^3 +... $$
But how to show that
$$ 1 + x + x^2 +x^3 +...= \frac{1}{1-x} $$
 [/INST] You have following expression-
$ 1 + x + x^2 +x^3 +...$
Now multiplying numerator and denominator by $x-1$, provided $x\neq1$, we have
$\frac{(1-x)( 1 + x + x^2 +x^3 +...)}{1-x}=\frac{(1+x+x^2...)-(x+x^2+x^3)}{1-x}=\frac{1}{1-x}$
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