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The Bellman optimality equation for the state-value function:^[1]

$$V^*(s) = \max_a \sum_{s'} P(s' \mid s, a)\left[R(s, a, s') + \gamma V^*(s')\right]$$

The action-value form:

$$Q^*(s,a) = \sum_{s'} P(s' \mid s, a)\left[R(s, a, s') + \gamma \max_{a'} Q^*(s', a')\right]$$

The discounted return:

$$G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + \cdots = \sum_{k=0}^{\infty} \gamma^k R_{t+k+1}$$

A deliberately wide equation to test horizontal scrolling on narrow screens:

$$\mathrm{Attention}(Q, K, V) = \mathrm{softmax}\!\left(\frac{QK^\top}{\sqrt{d_k}}\right)V \qquad \mathcal{L}(\theta) = -\mathbb{E}_{(x,y)\sim\mathcal{D}}\left[\sum_{t=1}^{T} \log p_\theta(y_t \mid y_{<t}, x)\right] + \lambda \lVert \theta \rVert_2^2$$

Inline math

The discount factor $\gamma \in [0, 1)$ controls how much the agent values future rewards, and the policy $\pi_\theta(a \mid s)$ is parameterized by $\theta$. TD error is $\delta_t = r_{t+1} + \gamma V(s_{t+1}) - V(s_t)$ inside a sentence.^[2]

Things that must NOT become math

OpenAI raised $6 billion in one round and later $250 million in another; Anthropic raised $8 billion from Amazon. A single $ sign stays text.

Inline code with dollars: price = "$$100" and echo $$PATH.

# Code fence: dollars stay literal
cost = "$$not math$$"
total = price * 2  # $50

Scroll sideways for more →

Citation marker near math: the value $V^\pi(s)$ was introduced earlier.^[1] See reinforcement learning.

References

1. Sutton, R. S., & Barto, A. G. (2018). Reinforcement Learning: An Introduction (2nd ed.). MIT Press. http://incompleteideas.net/book/the-book-2nd.html ↩
2. Bellman, R. (1957). Dynamic Programming. Princeton University Press. ↩