For a summation where the upper limit is not pre-defined, is $T=\infty$ meaningful?

Question

I have a function like

$$G_t = \sum_{t=0}^T \gamma^t R_t$$

I'm trying to save space in the document that I'm writing. I want to comment that there are two cases, $T \in \Bbb N$ or $T = \infty$. But I know that it's improper to say that something "equals" infinity because infinity is not a number.

Is it appropriate to say $T\rightarrow \infty$? Or is that also misleading?

Answer 1

I believe you intend to say $G_T$ rather than $G_t$ on th eleft hand side.

When we write $T=\infty$ in the upper index of the summation, it means $\lim_{T \to \infty} \sum_{t=0}^T \gamma^TR_t$.

This notation seems quite common in the readings on reinforcement learning.