Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (-4)ⁿ/n!

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Identify the general term of the sequence: \(a_n = \frac{(-4)^n}{n!}\).

Recall that \(n!\) (n factorial) grows much faster than any exponential function \(c^n\) as \(n\) approaches infinity.

To determine convergence, analyze the limit \(\lim_{n \to \infty} \frac{(-4)^n}{n!}\).

Since the factorial in the denominator grows faster than the exponential in the numerator, the terms \(a_n\) approach zero as \(n\) becomes very large.

Conclude that the sequence converges and its limit is \(0\).

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주요 개념

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Sequence Convergence

A sequence converges if its terms approach a specific finite value as n approaches infinity. Determining convergence involves analyzing the behavior of the nth term and checking if the limit exists and is finite.

Factorials and Growth Rates

Factorials (n!) grow much faster than exponential functions like (-4)^n. Understanding the relative growth rates helps in evaluating limits involving factorials and powers, often leading to convergence when factorials dominate.

Limit of a Sequence

The limit of a sequence aₙ is the value that aₙ approaches as n becomes very large. Calculating this limit often involves applying limit laws, comparing growth rates, or using known limits of standard sequences.

Determining Convergence of SequencesWhich of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.aₙ = (-4)ⁿ/n!