Formal Power Series

Let be a Ring.
We define to be the ring of power series, with elements

for any sequence
and with operations defined as:

where should be viewed as just a shorthand, and not an actual sum.

We will write to mean when
We also define for the smallest such that .

Series in a Ring
Formal Infinite Sum
Formal Infinite Product
Formal Composition
Formal Inverse
Formal Derivative
Formal Maclaurin
Formal Exponential
Formal Logarithm
Formal Binomial Power
Formal Binomial Theorem
Rogers-Ramanujan Identities