Modifying formula for finite sum with weight decay to limit decay to a minimum Let's suppose I have a sequence of non-negative integers $$x_0, x_1,...$$ Suppose $x_t$ represent the number of customers I acquire in period $t$.
Let $0< \beta < 1$
and
$$s_t = \sum_{i=0}^{t} \beta^{t-i} x_i $$
where $s_t$ is my total customers by period $t$
In the limit
$$\underset{t\rightarrow \infty}{\lim} \beta^{t-i} x_i = 0$$
Intuitively this means the decay term eventually drives the number of customers I have from period $t$ to 0 eventually. Makes sense. Fewer customers over time would stay from older periods.
But not all the customers should leave. That's not realistic for a decent business.
I want to change my formula so that
$$\underset{t\rightarrow \infty}{\lim} \beta^{t-i} x_i = c_i$$
where $c_{i}$ is essentially some minimum number of customers I've retained permanently. If it really helps, you could assume $c_{i}$ is the same for every period (e.g. $c$)  but I would prefer not to make that assumption.
How can I modify my formula for $s_t$ to accomplish that?
Keep in mind the following constraints.
Constraint 1:
$$\sum_{i=0}^{t} x_i \geq s_t$$
That is, you can't have more customers than there were for each period.
Constraint 2:
$$c_t \leq x_t$$
The number of retained customers in period $t$ must be less than or equal to the number customers acquired
 A: I question your assumptions (though I will show you how to go along with them if you insist). I think it's perfectly plausible that all the customers from period $i$ will eventually leave. If this does not match your intuition, then this is only because $t \to \infty$ is just a long time away.
But we can certainly imagine that there are short-term customers (whose number drops off quickly) and long-term customers (whose number drops off very slowly). We could in principle have any number of types of customers, but with too many types there is the danger of overfitting.
The simplest way to split customers into types is to assume a parameter $p \in [0,1]$ such that out of the $x_i$ new customers from period $i$:

*

*$p x_i$ are short-term customers (with decay rate $\beta$), and

*the remaining $(1-p)x_i$ are long-term customers (with a decay rate $\gamma$ much closer to $1$ than $\beta$ is).

In principle, $p$ could also vary with time according to some sequence, but that's not necessary.
At time $t$, only $\beta^{t-i}(px_i)$ of the short-term customers are left, but $\gamma^{t-i}((1-p)x_i)$ of the long-term customers are left. Altogether, we can write
$$
    s_t = \sum_{i=0}^t \big(\beta^{t-i}p + \gamma^{t-i}(1-p)\big) x_i.
$$
If you really wanted to model permanent customers, you could set $\gamma$ equal to $1$. Then, as $t \to \infty$, $\beta^{t-i}(p x_i) \to 0$, but the $(1-p) x_i$ permanent customers stay forever.
