<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] $$1+x+x^2+x^3+\dots$$
$$=(1+x)(1+x^2+x^4+\dots)$$
$$=(1+x)(1+x^2)(1+x^4+\dots)$$
$$=(1+x)(1+x^2)(1+x^4)\dots$$
$$=\frac{1-x^2}{1-x}\frac{1-x^4}{1-x^2}\frac{1-x^8}{1-x^4}\dots$$
$$=\frac{1}{1-x}$$
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