Domanda del libro di testo
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)(1 − cos(1 / k))²
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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.37
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²
Guida verificata passo dopo passo1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{2^k}{3^{k+2}} \). Notice that the series involves terms with exponents depending on \(k\).
Rewrite the general term to separate the powers of 3: \( \frac{2^k}{3^{k+2}} = \frac{2^k}{3^k \cdot 3^2} = \frac{1}{3^2} \cdot \frac{2^k}{3^k} = \frac{1}{9} \cdot \left( \frac{2}{3} \right)^k \).
Recognize that the series is a geometric series with the first term \( a = \frac{1}{9} \cdot \left( \frac{2}{3} \right)^1 = \frac{2}{27} \) and common ratio \( r = \frac{2}{3} \).
Check the convergence of the geometric series by verifying if \( |r| < 1 \). Since \( \frac{2}{3} < 1 \), the series converges.
Use the formula for the sum of an infinite geometric series starting at \( k=1 \): \[ S = a \cdot \frac{1}{1 - r} \], where \( a \) is the first term and \( r \) is the common ratio. Substitute the values to express the sum.
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Infinite Geometric Series
An infinite geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It converges if the absolute value of the ratio is less than 1, and its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
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Percorso guidato
06:00
Geometric Series
Convergence and Divergence of Series
A series converges if the sum of its infinite terms approaches a finite limit; otherwise, it diverges. For geometric series, convergence depends on the common ratio's magnitude. Understanding convergence is essential to determine whether the series sum exists or not.
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Percorso guidato
06:52Convergence of an Infinite Series
Manipulating Series Terms
Simplifying the general term of a series often involves algebraic manipulation, such as factoring exponents or rewriting terms to identify the first term and common ratio. This step is crucial to apply known formulas and test for convergence effectively.
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Percorso guidato
06:00Geometric Series
Pratica correlata
Domanda del libro di testo
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)tanh(k)
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Domanda del libro di testo
Building a tunnel — first scenario
A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.
a.How far does the crew dig in 10 weeks? 20 weeks? N weeks?
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Domanda del libro di testo
Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.
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Domanda del libro di testo
Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.
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Domanda del libro di testo
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷
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