Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problem 10.PE.18

Chapter 10, Problem 10.PE.18

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

Textbook Question

Convergent Series

Find the sums of the series in Exercises 19–24.

∑ (from n = 2 to ∞) -2/[n(n+1)]

Textbook Question

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) xⁿ/nⁿ

Textbook Question

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 = 1 + (0.9)ⁿ

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Textbook Question

Maclaurin Series

Find Taylor series at x = 0 for the functions in Exercises 63–70.

cos (x³/√5)