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
$$ 1 + x + x^2 +x^3 +...= \frac{1}{1-x} $$

We start with the simple geometric series
$$\sum_{k=0}^n x^k = \dfrac{1-x^{n+1}}{1-x}$$
Now assume that $|x|<1$.
With this assumption in mind, taking the limit $n \to \infty$ gives,
$$\sum_{k=0}^{\infty} x^k = \dfrac{1}{1-x} \ , \ |x|<1$$
The upper $x$ term cancelled out due our assumption.
