Why are these equations equal? I have racked my brain to death trying to understand how these two equations are equal:
$$\frac{1}{1-q} = 1 + q + q^2 + q^3 + \cdots$$
as found in http://www.math.dartmouth.edu/archive/m68f03/public_html/partitions.pdf
From what I understand if I substitute $5$ for $q$ the answer:
$$\frac{1}{1-5} = -\frac{1}{4}$$
is much different than:
$$1+5+25+125+\cdots$$
Any help understanding what is going on would be GREATLY appreciated. Thank you all for your time

$S=1+q+q^2+\dots$
For any $k\in \mathbf{N}$,
Let $S_k=1+q+q^2+\dots+q^k$,
Then, $qS_k=q+q^2+q^3+\dots+q^{k+1}$.
So, $(1-q)S_k=S_k-qS_k=1+q+q^2+\dots+q^k-(q+q^2+q^3+\dots+q^{k+1})=1-q^{k+1}$
If $|q|<1$, then $q^k \longrightarrow 0$ as $k \longrightarrow \infty$, so,
$$S=\lim_{k \to \infty}S_k=\lim_{k \to \infty}\frac{1-q^{k+1}}{1-q} =\frac{1}{1-q}$$
Note, however, that the function $$f(x)=\frac{1}{1-x}$$ is valid for all $x \neq 1$. But, $f(x) = 1+x+x^2+\dots$ only for $|x| <1$.
