Domanda del libro di testo
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
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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 10.4.27
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 = 1 to ∞) k^(1/k)
Guida verificata passo dopo passo1
First, identify the general term of the series: \(a_k = k^{1/k}\).
Apply the Divergence Test by finding the limit of \(a_k\) as \(k\) approaches infinity: compute \(\lim_{k \to \infty} k^{1/k}\).
Rewrite the term inside the limit using exponentials and logarithms: \(k^{1/k} = e^{(1/k) \ln(k)}\).
Evaluate the limit of the exponent: \(\lim_{k \to \infty} \frac{\ln(k)}{k}\), which approaches 0, so the limit of \(a_k\) is \(e^0 = 1\).
Since the limit of \(a_k\) is 1 (not zero), by the Divergence Test, the series \(\sum_{k=1}^\infty k^{1/k}\) diverges.
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Divergence Test
The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. It is a quick initial check to determine if a series can possibly converge.
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Percorso guidato
05:44
Divergence Test (nth Term Test)
Integral Test
The Integral Test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of the function from 1 to infinity converges, then the series converges; if the integral diverges, so does the series.
Video consigliato:
Percorso guidato
07:25Integral Test
p-series Test
The p-series Test applies to series of the form ∑ 1/k^p. Such a series converges if and only if p > 1, and diverges otherwise. It is useful for comparing or identifying the behavior of series with terms involving powers of k.
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Percorso guidato
04:30P-Series and Harmonic Series
Pratica correlata
Domanda del libro di testo
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (2k + 1)! / (k!)²
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Domanda del libro di testo
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / lnk
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Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)
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Domanda del libro di testo
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.
61. ∑ (k = 1 to ∞) ln((k + 1) / k)
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Domanda del libro di testo
Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?
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