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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.3.87e
Chapter 10, Problem 10.3.87e

87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.


e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.

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Identify the general term of the series: the series is given by \( \sum_{k=1}^{\infty} \left( \frac{\pi}{e} \right)^{-k} \). This can be rewritten as \( \sum_{k=1}^{\infty} \left( \frac{e}{\pi} \right)^k \) because raising to the power \(-k\) is the same as taking the reciprocal and raising to the positive power \(k\).
Recognize that this is a geometric series with the first term \( a = \left( \frac{e}{\pi} \right)^1 = \frac{e}{\pi} \) and common ratio \( r = \frac{e}{\pi} \).
Recall the convergence criterion for a geometric series: a geometric series \( \sum a r^{k-1} \) converges if and only if \( |r| < 1 \).
Evaluate the absolute value of the common ratio: since \( e \approx 2.718 \) and \( \pi \approx 3.1415 \), \( \left| \frac{e}{\pi} \right| < 1 \).
Conclude that because \( |r| < 1 \), the series \( \sum_{k=1}^{\infty} \left( \frac{e}{\pi} \right)^k \) is a convergent geometric series.

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Key Concepts

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

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio r. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding the structure helps identify if a series fits this pattern.
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Geometric Series

Convergence of Geometric Series

A geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. When this condition holds, the infinite sum approaches a finite limit given by a/(1-r). If |r| ≥ 1, the series diverges.
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Evaluating the Common Ratio

To determine convergence, it is essential to correctly identify and evaluate the common ratio r in the series. In this problem, the term (π/e)^(-k) can be rewritten as (e/π)^k, so the ratio is e/π. Comparing |e/π| to 1 determines if the series converges.
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Graphs of Common Functions
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.

Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

f.If the sequence {aₙ} diverges, then the sequence {0.000001aₙ} diverges.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


d.If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and

{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},

then limₙ→∞aₙ = limₙ→∞bₙ.

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