Domanda del libro di testo
Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.
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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.R.11b
b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?
Guida verificata passo dopo passo1
Identify the general term of the series: \(a_k = \frac{k}{k+1}\).
Examine the behavior of the terms \(a_k\) as \(k\) approaches infinity by finding \(\lim_{k \to \infty} a_k\).
Calculate the limit: \(\lim_{k \to \infty} \frac{k}{k+1} = \lim_{k \to \infty} \frac{k}{k(1 + \frac{1}{k})} = \lim_{k \to \infty} \frac{1}{1 + \frac{1}{k}}\).
Since \(\lim_{k \to \infty} a_k = 1 \neq 0\), recall the necessary condition for series convergence: if the terms do not approach zero, the series cannot converge.
Conclude that because the terms do not approach zero, the series \(\sum_{k=1}^\infty \frac{k}{k+1}\) diverges.
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Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Convergence means the series approaches a finite limit as more terms are added. Determining convergence involves analyzing the behavior of the partial sums or applying convergence tests.
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Percorso guidato
06:52
Convergence of an Infinite Series
Term Test for Divergence
If the terms of a series do not approach zero as k approaches infinity, the series cannot converge. This is a quick initial test to rule out convergence by examining the limit of the general term.
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Percorso guidato
05:44Divergence Test (nth Term Test)
Behavior of the General Term k/(k+1)
The term k/(k+1) simplifies to 1 - 1/(k+1), which approaches 1 as k grows large. Since the terms do not approach zero, the series ∑ k/(k+1) diverges by the term test.
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Percorso guidato
06:11Introduction to Riemann Sums
Pratica correlata
Domanda del libro di testo
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ
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Domanda del libro di testo
25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.
b.Determine the limit of each sequence.
25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80
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Domanda del libro di testo
{Use of Tech} Error in a finite alternating sum
How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?
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Domanda del libro di testo
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ
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Domanda del libro di testo
Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.
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