Discussion
The type of the sequence \(a_{n} = \left(\frac{n}{n - 1}\right)^{3}\) is
Answer: C
The sequence is converging.
\(\frac{n}{n - 1} = 1 + \frac{1}{n - 1}\), which tends to 1 as \(n \to \infty\), so \(a_{n} = \left(\frac{n}{n - 1}\right)^{3} \to 1\). The terms decrease steadily toward this limit (\(a_{2} = 8\), \(a_{3} = 3.375\), \(a_{4} \approx 2.37\), ...), so the sequence is monotonic and convergent.
It is not oscillatory, since the terms never alternate, and not divergent, since it has a finite limit. For \(n \ge 2\) it is also bounded (between 1 and 8), but boundedness is a property every convergent sequence has; the key takes 'converging' as the precise description of its type.
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