= Solution
For the symmetric <simple random walk on the integer line>, odd return times are impossible. A return after $2m$ steps requires exactly $m$ positive increments, so
$$
p_{00}^{(2m)}=\binom{2m}{m}2^{-2m}\sim\frac1{\sqrt{\pi m}}
$$
by <Stirling's approximation>. The resulting sum diverges by the <p-series> criterion. The <recurrence criterion by return probabilities> therefore makes zero a <recurrent state>. The <simple random walk> is an <irreducible Markov chain> on $\mathbb Z$, so \b[the symmetric chain is recurrent].
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