Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain such that $X_{0}=i$. Prove that

$$\sum_{n=0}^{\infty} \mathbb{P}_{i}\left(X_{n}=i\right)=\frac{1}{\mathbb{P}_{i}\left(X_{n} \neq i \text { for all } n \geqslant 1\right)}$$

where $1 / 0=+\infty$. [You may use the strong Markov property without proof.]