= Solution
Let $R$ be the number of returns to $v$ after time $0$. Since $v$ is a <recurrent state>, the probability of another return after each visit is one. By the stated <geometric distribution> description, $\mathbb P_v(R\geq k)=1$ for every positive integer $k$. Thus $R=\infty$ almost surely and $\mathbb E_vR=\infty$.
On the other hand, writing each visit as an <indicator random variable> and applying the <monotone convergence theorem> to the partial sums gives
$$
1+\mathbb E_vR
=\mathbb E_v\left[\sum_{n=0}^{\infty}\boldsymbol1_{\{X_n=v\}}\right]
=\sum_{n=0}^{\infty}\mathbb P_v(X_n=v)
=\sum_{n=0}^{\infty}p_{vv}(n).
$$
The left side is infinite, proving the <recurrence criterion by return probabilities> in the required direction:
$$
\boxed{\sum_{n=0}^{\infty}p_{vv}(n)=\infty}.
$$
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