§3, discounted case (p. 446) — the discounted cost of a sigma in closed form (indices corrected)

Let $0<\beta<1$ and let $P$ be a sigma from $s_0$ in a deterministic Markov decision process: a walk $w_0=s_0,\dots,w_N$ along decisions $e_0,\dots,e_{N-1}$ with distinct nodes $w_0,\dots,w_{N-1}$ and $w_N=w_j$ for some $j<N$. Write $c_t=c(w_t,e_t)$ for the cost of its $t$-th arc and $l=N-j\ge 1$ for the length of its cycle. Then the discounted cost of the infinite walk that follows $P$ and repeats the cycle forever is

This is the formula the paper uses to evaluate a sigma; it turns the infinite discounted sum into a finite expression in the prefix cost and the cycle cost.

Formalization Note The paper prints $c(P)=\sum_{i=0}^{k}c(u_j,u_{j+1})\beta^i+\frac{\beta^{k+1}}{1-\beta^l}\sum_{j=1}^{l}c(v_i,v_{i+1 \bmod l})\beta^j$, with mixed summation indices and a factor $\beta^j$ in the cycle sum that is one power of $\beta$ too many for the sentence that follows it ("the discounted cost of a sigma coincides with the discounted cost of an infinite path that follows the sigma and repeats the cycle forever"). We state the corrected identity, which agrees with that sentence: with the paper's $j=k+1$, the cycle's $t$-th arc ($t=0,\dots,l-1$) carries $\beta^{t}$, i.e. the paper's $\beta^{j-1}$ for $j=1,\dots,l$, and $u_{k+1}=v_1$. The left side is the definition of the sigma's cost (an infinite series); $0<\beta<1$ is the paper's $\beta\in(0,1)$.