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.4

Chapter 10, Problem 10.PE.4

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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Identify the general term of the sequence: \(a_n = 1 + (0.9)^n\).

Recall that a sequence converges if its terms approach a finite limit as \(n\) approaches infinity.

Analyze the behavior of the term \((0.9)^n\) as \(n \to \infty\). Since \(0.9\) is between \(-1\) and \(1\), \((0.9)^n\) approaches \(0\).

Use the limit laws to find the limit of \(a_n\): \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1 + (0.9)^n\right) = 1 + 0\).

Conclude that the sequence converges and its limit is \(1\).

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Key Concepts

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

Definition of Sequence Convergence

A sequence converges if its terms approach a specific finite value, called the limit, as n approaches infinity. Formally, a sequence {aₙ} converges to L if for every small positive number ε, there exists an N such that for all n > N, |aₙ - L| < ε.

Limits of Exponential Terms

When a sequence includes terms like (r)ⁿ where |r| < 1, these terms approach zero as n becomes very large. This property helps determine the limit of sequences involving exponential decay factors, such as (0.9)ⁿ approaching 0.

Evaluating Limits of Sequences

To find the limit of a sequence, analyze each component separately and use limit laws. For example, in aₙ = 1 + (0.9)ⁿ, since (0.9)ⁿ → 0, the sequence converges to 1 + 0 = 1.

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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

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = ln(cos x)

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 = (-4)ⁿ/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)