Convergence of Sequence an Fraction n Factorial

Determine if the sequence $a_n = \frac{n-1}{n!}$ converges or diverges as $n \to \infty$.

Sequences are an essential part of mathematical analysis and are closely related to series. Understanding whether a sequence converges or diverges requires analyzing its behavior as the variable approaches infinity. In this problem, we are examining the sequence $a_n = \frac{n-1}{n!}$, which involves factorials — a fundamental concept with significant growth properties. Factorials grow very rapidly compared to polynomials or even exponential functions, which can greatly impact the behavior of sequences and series associated with them.

When determining convergence, a common approach is to consider the limit of the sequence as n approaches infinity. For this type of sequence, considering the comparative growth rates of the numerator and the denominator is crucial. The numerator, n-1, grows linearly, while the denominator, n!, grows factorially. This stark difference in growth rates is a strong indicator of convergence behavior, as the denominator will dominate the numerator as n becomes large.

Ultimately, understanding the convergence of sequences involves applying various tests and employing intuitive understandings of underlying mathematical properties, such as growth rates and limits. The study of sequences not only helps in foundational mathematics but also plays an important role in fields such as calculus, real analysis, and beyond.

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