Ch. 10 - Infinite Sequences and Series

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Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problema 10.PE.4

Capitolo 10, Problema 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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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.

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Introduction to Sequences

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.

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Exponential Functions

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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Introduction to Sequences

Pratica correlata

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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 ∞) (x + 4)ⁿ/(n3ⁿ)

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Convergent Series

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

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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

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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 ∞) (csch n)xⁿ

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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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Maclaurin Series

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

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