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

The sum on the RHS of your equation is an infinite geometric series, whose sum is $\frac {1}{1-q}$ only if $|q|$ is less than 1. Your example of $q=5$ is therefore not valid.
