<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] $$ \text{Let }S_{n}=1+x+x^2...+x^{n-1}$$
$$\implies xS_{n}=x+x^2....+x^{n}$$
Subtracting both equations,
$$S_{n}(1-x)=1-x^{n}$$
$$\implies S_{n}=\frac{1-x^n}{1-x}$$
Since it is an infinite series, $n\to\infty$ and it converges only when $|x|<1$. When $n\to\infty,x^n\to0$
$$\implies s_{\infty}=\frac{1-0}{1-x}=\frac{1}{1-x} $$
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