# 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?

• The case of differentiation is a good one because things can get a bit weird once you start looking at distributions. Take for instance the Heaviside step function. It is not differentiable in the usual sense at $0$ however distributionally, it has a Dirac delta as a derivative. So you could say "formally $H'(x) = \delta(x)$" or something along those lines. – Cameron Williams Jul 27 '13 at 21:13

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.
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.
The formal power series over a commutative ring are basically just the sequences in that ring, with addition and multiplication defined to match. So $\sum_{n\in\Bbb N}a_n x^n$ is basically equivalent to the sequence $(a_n)_{n\in \Bbb N}$, but there are rules for addition, subtraction, and multiplication of the former: $$\sum_{n\in \Bbb N}a_n x^n + \sum_{n\in N}b_n x^n := \sum_{n\in\Bbb N}(a_n+b_n)x^n,$$ $$-\sum_{n\in\Bbb N}a_n x^n = \sum_{n\in\Bbb N}(-a_n)x^n,$$ and $$\Bigl(\sum_{n\in \Bbb N}a_nx^n\Bigr)\Bigl(\sum_{m\in\Bbb N}b_nx^m\Bigr):=\sum_{n\in \Bbb N}\Bigl(\sum_{j+k=n}a_jb_k\Bigr)x^n.$$