Exercise:
If the $y \in \mathbb X$ is a transient state, show that for all $x \in \mathbb X$, it is: $$\mathbb E_x \left[V(y)\right] = \frac{\mathbb P_x\left[T_y < +\infty\right]}{\mathbb P_y\left[T_y^+ < + \infty\right]}.$$
First, clarifying the symbols.
As $V(y)$ we denote the number of lifetime arrivals at y of the given Markov Chain:
$$V(y) = \sum_{n=0}^\infty \mathbf 1_{\{X_n = y\}}.$$
As $T_y$ we denote the time of the first arrival at y (hitting time): $$T_y = \inf\left\{k \geq 0 : X_k = y \right\}.$$
As $T^+_y$ we denote the time of the first return to y (first hit time): $$T_y^+ = \inf\left\{ k \geq 1 : X_k = y \right\}.$$
So, we are interested in calculating the expected value:
$$\mathbb E\left[ V(s) \big| X_0 = x\right] = \mathbb E_x \left[V(y)\right] = \mathbb E_x \left[ \sum_{n=0}^\infty 1_{\{X_n = y\}} \right]= \sum_{n=0}^\infty \mathbb E_x \left[1_{\{X_n = y\}}\right] = \sum_{n=0}^\infty \mathbb P_x \left[X_k = y\right]$$
Now, if my in-between steps are correct, how does one arrive at the expression needed?