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