The decimal point is ambiguous

Many numbers can be symbolically written in more than one way. For example :

The decimal point notation also has its own problems. For example, consider the following perfectly valid identity :

with the series of s repeating ad infinitum. There are many ways to prove the above. Here is one.

The right-hand side can be written in a form involving a geometric progression :

The series converges in . Its value can thus be computed using the usual formula

with , , which, in our case gives

Which proves the identity.