Time to make "stationary" precise. A series is weakly (covariance) stationary if three things hold for all t: (1) its mean E[x_t] is constant, not a function of t; (2) its variance is constant, finite, and also not a function of t; (3) its autocovariance Cov(x_t, x_{t-k}) depends only on the lag k, never on t itself. Every AR(1) with |φ|<1 and every finite MA process satisfies all three. The random walk, x_t = x_{t-1} + ε_t (AR(1) with φ = 1 exactly, a unit root), satisfies none of them: its variance is Var(x_t) = t·σ², which grows without bound as t grows, a fact you can verify by simulating twenty thousand random-walk paths and checking the sample variance at several horizons against t·σ² directly (with σ = 1.3: at t=10 the simulated variance is 16.97 against a theoretical 16.90; at t=100, 168.43 against 169.00; at t=200, 338.63 against 338.00, the match tightens as the sample of paths grows, exactly as the law of large numbers promises). A variance that depends on how far you've walked is the mathematical fingerprint of a unit root, and it is why a random walk can produce long stretches that look, to the naked eye, exactly like a deterministic trend (Worked Example 3 builds one on purpose).
Why this matters beyond aesthetics: when x_t has a unit root, the ordinary-least-squares machinery from QM1.02 stops behaving normally. The usual t-statistics no longer follow a Student-t distribution even asymptotically; they follow a non-standard distribution, skewed so that "large" t-statistics happen far more often under the null than the ordinary tables suggest. That single fact is the root cause of the spurious-regression phenomenon (see cointegration and spurious regression), and it is why you cannot simply eyeball a trending series, fit a trend line, and trust the fit.
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How does the Augmented Dickey-Fuller test work?
The standard detection tool is the (Augmented) Dickey-Fuller test (Dickey & Fuller, 1979). Rewritten as a testable regression, Δx_t = α + γ·x_{t-1} + Σδᵢ·Δx_{t-i} + ε_t (the lagged-difference terms are the "augmentation," soaking up serial correlation in the errors so the test stays valid), the null hypothesis is γ = 0 (a unit root is present) against the alternative γ < 0, the series is stationary. Because this t-statistic's null distribution is non-standard, its critical values are not the ordinary t-table, they come from Dickey and Fuller's own tabulation (implemented via MacKinnon's response-surface approximations) and are more negative, meaning you need more evidence than usual to declare a series stationary. Reading the test: if the ADF statistic is more negative than the critical value, reject the null (stationary); if less negative (closer to zero or positive), fail to reject (a unit root is not ruled out). Worked Example 3 runs this on a price level that visibly climbs for three years, and on its daily changes, with genuinely surprising results.