1. The problem asks: What does it mean for a sequence $x_n$ to be divergent? 2. In mathematics, a sequence $x_n$ is said to be divergent if it does not converge to a finite limit as $n$ approaches infinity. 3. More formally, a sequence $x_n$ diverges if there is no real number $L$ such that $$\lim_{n \to \infty} x_n = L.$$ This means the terms of the sequence do not get arbitrarily close to any single finite value. 4. Important rules: - If $x_n$ grows without bound (e.g., $x_n \to \infty$ or $x_n \to -\infty$), it is divergent. - If $x_n$ oscillates between values without settling, it is divergent. 5. Examples: - The sequence $x_n = n$ diverges because it increases without bound. - The sequence $x_n = (-1)^n$ diverges because it oscillates between $1$ and $-1$. 6. In summary, divergence means the sequence does not approach any single finite number as $n$ becomes very large.