Formal (series/sum/derivative...) I have come across a lot of cases where terms such as formal sum rather than simply sum is used, similarly in case of derivatives/infinite series/power series.
As I understand in case of series/sum, the term formal is used when the notion of convergence is not clear.
I would appreciate any precise definition or explanation of where formal is used. I also cannot get how it relates to derivatives. Also, where else is "formal" used?
 A: A formal sum is where we write something using a $+$ symbol, or other way normally used for sums, even when there may be no actual operation defined.  
An example.  Someone may have defined a quaternion as a formal sum of a scalar and a (3-dimensional) vector, for example.  Before this definition, at least in that book, there was no defined "sum" of a scalar plus a vector.  
After that definition, our author may tell us what is means for two quaternions to be equal
$$
\lambda + \mathbf{x} = \mu + \mathbf{y} \quad \Longleftrightarrow \quad\text{??}
$$
how to add quaterntions
$$
(\lambda + \mathbf{x}) + (\mu + \mathbf{y}) = \text{??}
$$
how to multiply quaterntions
$$
(\lambda + \mathbf{x}) \cdot (\mu + \mathbf{y}) = \text{??}
$$
and so on.
A: The formal power series over a commutative ring are basically just the sequences in that ring, with addition and multiplication defined to match. So the formal power series $\sum_{n\in\Bbb N}a_n x^n$ corresponds to the sequence $(a_n)_{n\in \Bbb N}$. There are rules for addition, subtraction, and multiplication of formal power series, specifically:
$$\sum_{n\in \Bbb N}a_n x^n + \sum_{n\in N}b_n x^n := \sum_{n\in\Bbb N}(a_n+b_n)x^n,$$
$$-\sum_{n\in\Bbb N}a_n x^n = \sum_{n\in\Bbb N}(-a_n)x^n,$$
and
$$\Bigl(\sum_{n\in \Bbb N}a_nx^n\Bigr)\Bigl(\sum_{m\in\Bbb N}b_nx^m\Bigr):=\sum_{n\in \Bbb N}\Bigl(\sum_{j+k=n}a_jb_k\Bigr)x^n.$$
