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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.1.73
Capitolo 10, Problema 10.1.73

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)!

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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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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.
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Introduction to Sequences

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.
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Factorials

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.
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Introduction to Sequences
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Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

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Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

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Absolute and Conditional Convergence

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