= Solution
Series 1 wanders over a changing level rather than fluctuating around a stable local mean. Its sample <autocorrelation function> is strongly positive and decreases very slowly. This is the usual diagnostic evidence for an ordinary <unit root>: an autoregressive polynomial containing $1-z$, with a zero at $z=1$, and a stationary model after first <differencing>. The plots support an integrated model, rather than specifying the number of its remaining stationary autoregressive or moving-average terms.
Series 2 has a pronounced oscillation with period about six observations. Its sample <autocorrelation> alternates between large positive and negative values with little damping: approximately positive at multiples of six and negative halfway between. Together with the changing amplitude, this suggests a conjugate pair of unit-circle zeros near
$$
\boxed{z=e^{\pm i\pi/3}.}
$$
The associated real autoregressive factor is $1-2\cos(\pi/3)z+z^2=1-z+z^2$. A targeted filter $1-B+B^2$ removes this pair; the broader <seasonal difference operator> $1-B^6$ also contains it but introduces additional <differencing> factors. This is the <oscillatory unit-root diagnosis from an undamped sample autocorrelation>.
Thus \b[Series 1 suggests a zero at 1; Series 2 suggests a conjugate pair on the unit circle at a seasonal frequency]. These are model diagnoses, not deductions of exact roots from a finite sample. A stationary model very close to a <unit root> can look similar, and an undamped periodic <covariance> can also arise from a stationary random sinusoid. The figure does not identify exact orders or prove nonstationarity by itself.
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