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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.4.47e
Capitolo 10, Problema 10.4.47e

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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Recall the p-series test: The series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if and only if \( p > 1 \).
Given that \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges, it implies that \( p > 1 \).
Now consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^{p + 0.001}} \). Since \( p + 0.001 > p > 1 \), this new exponent is also greater than 1.
By the p-series test again, since the exponent \( p + 0.001 \) is greater than 1, the series \( \sum_{k=1}^{\infty} \frac{1}{k^{p + 0.001}} \) also converges.
Therefore, the statement is true because increasing the exponent in a convergent p-series to a slightly larger value still results in a convergent series.

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p-Series Test for Convergence

The p-series test states that the series ∑ 1/k^p converges if and only if p > 1. This test helps determine whether a series with terms involving powers of k converges or diverges based on the value of p.
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Percorso guidato
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P-Series and Harmonic Series

Comparison Test for Series

The comparison test allows us to determine convergence by comparing a given series to another series with known behavior. If a series with larger terms converges, then a series with smaller terms also converges, and vice versa.
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Percorso guidato
09:25
Direct Comparison Test

Effect of Small Changes in the Exponent on Convergence

Slightly increasing the exponent p in a p-series (e.g., from p to p + 0.001) affects convergence because the series terms decrease faster. If the original series converges, increasing p will also result in convergence, but if it diverges, the new series may still diverge.
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5:50
Power Rules
Pratica correlata
Domanda del libro di testo

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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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.

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

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Domanda del libro di testo

87. Explain why or why not

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


f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ 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. 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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