Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (2n + 2)! / (2n − 1)!

검증된 단계별 안내

Identify the given sequence: \(a_n = \frac{(2n + 2)!}{(2n - 1)!}\).

Recall that factorial notation means the product of all positive integers up to that number, so \((2n + 2)!\) expands to \((2n + 2)(2n + 1)(2n)(2n - 1)!\).

Rewrite the sequence by factoring out \((2n - 1)!\) from the numerator to simplify the expression: \(a_n = \frac{(2n + 2)(2n + 1)(2n)(2n - 1)!}{(2n - 1)!}\).

Cancel the common factorial term \((2n - 1)!\) in numerator and denominator, leaving \(a_n = (2n + 2)(2n + 1)(2n)\).

Analyze the behavior of \(a_n\) as \(n \to \infty\): since it is a product of terms that grow without bound, the sequence diverges (does not converge to a finite limit).

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주요 개념

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Sequence Convergence and Divergence

A sequence converges if its terms approach a finite limit as n approaches infinity; otherwise, it diverges. Understanding this helps determine the behavior of {aₙ} by analyzing its limit or growth pattern.

Factorials and Their Growth Rates

Factorials (n!) grow very rapidly as n increases. Comparing factorial expressions like (2n+2)! and (2n−1)! requires understanding how factorial terms expand and dominate, which is crucial for evaluating the limit of the sequence.

Limit Evaluation Techniques for Sequences

Techniques such as simplifying factorial expressions, using ratio tests, or applying Stirling’s approximation help find limits of sequences involving factorials. These methods allow precise determination of whether the sequence converges and to what value.

Convergence and DivergenceWhich of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.aₙ = (2n + 2)! / (2n − 1)!