Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
41.∑ (k = 1 to ∞) 4 / 12ᵏ
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.8.45
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)
Guida verificata passo dopo passo1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{4}}{e^{k^{5}}} \). We want to determine if this infinite series converges.
Consider the general behavior of the terms \( a_k = \frac{k^{4}}{e^{k^{5}}} \). Notice that the denominator grows exponentially with respect to \( k^{5} \), while the numerator grows polynomially as \( k^{4} \).
Apply the Comparison Test or Limit Comparison Test by comparing \( a_k \) to a simpler series. Since exponential growth dominates polynomial growth, compare \( a_k \) to \( \frac{1}{e^{k^{5}}} \), which is a convergent series because its terms approach zero very rapidly.
Since \( \sum \frac{1}{e^{k^{5}}} \) converges (it is a series with terms decreasing faster than any geometric series), and \( \frac{k^{4}}{e^{k^{5}}} \leq C \cdot \frac{1}{e^{k^{5}}} \) for some constant \( C \) and sufficiently large \( k \), by the Comparison Test, the original series converges.
Conclude that the series \( \sum_{k=1}^{\infty} \frac{k^{4}}{e^{k^{5}}} \) converges absolutely due to the dominance of the exponential term in the denominator.
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Convergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to determine whether the series sums to a finite value or diverges to infinity.
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Percorso guidato
06:52
Convergence of an Infinite Series
Comparison Test
The Comparison Test involves comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps analyze series with complicated terms.
Video consigliato:
Percorso guidato
09:25Direct Comparison Test
Exponential vs. Polynomial Growth
Exponential functions grow faster than any polynomial function as the variable approaches infinity. In series terms, if the denominator grows exponentially and the numerator polynomially, the terms tend to zero rapidly, often ensuring convergence.
Video consigliato:
09:29Exponential Growth & Decay
Pratica correlata
Domanda del libro di testo
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b.Use analytical methods to find the limit of the sequence.
aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3
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Domanda del libro di testo
What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?
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Domanda del libro di testo
49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.
aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …
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Domanda del libro di testo
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln20n}
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Domanda del libro di testo
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{ⁿ√(e³ⁿ⁺⁴)}
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