What is the correct mathematical notation for something comprised of the sum of constituents n where n is infinite?

Question

I am trying to figure out what the correct mathematical notation would be for something like the following:

I want to describe that the value V of a company is equal to sum of parameters P at any given time t. The amount of parameters can in theory infinite. Understanding that it is impractical to determine an infinite amount of parameters, perhaps even impossible to describe all parameters I'm thinking about including a residual parameter to account for "unknowns".

What would be best way to describe such a relationship?

This is what I have come up with but I am unsure if this is mathematically bullet proof "proof-wise" and what the implications are of below formula.

$$ V_{a_t} = \sum_{i=1}^nP_{i_t} + e; \lim_{n\to\infty} n = \infty $$

E.g. should $e$ be included or is better to just leave it out? Hoping for some help as I am not an expert in the math field.

Answer 1

The way I would write this is $$V(t)=\sum\limits_{i=1}^{n} P_i (t)+\epsilon_n (t)$$

Where $\lim\limits_{n\to\infty} \epsilon_n (t)=0$ for all $t$

Answer 2

You can readily write $V(t)=\sum_{i\in I} P_i(t)$ where $I$ is an index set (which may be infinite). However, in the infinite case we do not really have a sum with all its nice properties any more (unless all but finitel many summands are zero):

• The sum may not be defined (converge) if "too many" of the $P_i(t)$ are "too large"
• Especially, if $I$ is uncountable and uncountably many $P_i(t)$ are nonzero, we are in serious trouble
• If the $P_i(t)$ have different signs then without a natural or implied order on $I$ the "sum" is not defined (i.e., different orders may result in differnt sums)