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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 10.5.9
Capitolo 10, Problema 10.5.9

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k² + 4)

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{k^2 + 4} \). We want to determine if this series converges.
Choose a comparison series that is simpler but behaves similarly for large \( k \). Since \( k^2 + 4 \) behaves like \( k^2 \) for large \( k \), consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a p-series with \( p = 2 \).
Recall that the p-series \( \sum \frac{1}{k^p} \) converges if \( p > 1 \). Since \( p = 2 > 1 \), the comparison series \( \sum \frac{1}{k^2} \) converges.
Apply the Limit Comparison Test by computing the limit \( L = \lim_{k \to \infty} \frac{\frac{1}{k^2 + 4}}{\frac{1}{k^2}} = \lim_{k \to \infty} \frac{k^2}{k^2 + 4} \).
Evaluate the limit \( L \). If \( L \) is a finite positive number, then both series either both converge or both diverge. Since the comparison series converges, this will imply the original series converges as well.

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Comparison Test

The Comparison Test determines the convergence of a series by comparing it to another series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges.
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Percorso guidato
09:25
Direct Comparison Test

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar term behavior.
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07:45
Limit Comparison Test

p-Series and Their Convergence

A p-series has the form ∑ 1/k^p and converges if p > 1, diverging otherwise. Recognizing that 1/(k² + 4) behaves like 1/k² for large k helps apply comparison tests effectively, since the p-series with p=2 is known to converge.
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P-Series and Harmonic Series
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13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{1 + cos(1⁄n)}

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23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 3 to ∞) 1 / (k − 2)⁴

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54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

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6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{1.00001ⁿ}

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13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(1 / n)¹⁄ⁿ}

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For what values of p does the series ∑ (k = 10 to ∞) 1 / kᵖ converge (initial index is 10)? For what values of p does it diverge?

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