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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.3.87e
Capitolo 10, Problema 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.

Guida verificata passo dopo passo
1
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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Concetti chiave

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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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Geometric Series

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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5:57
Graphs of Common Functions
Pratica correlata
Domanda del libro di testo

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.


41. ∑ (k = 1 to ∞) 1 / k⁶

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Domanda del libro di testo

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.

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Domanda del libro di testo

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.

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Domanda del libro di testo

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


d. The Ratio Test is always inconclusive when applied to ∑ aₖ, where aₖ is a nonzero rational function of k.

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Domanda del libro di testo

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."

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Domanda del libro di testo

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


d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².

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