Domanda del libro di testo
67–70. Formulas for sequences of partial sums Consider the following infinite series.
c.Make a conjecture for the value of the series.
∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]
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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 10.7.31c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.
Guida verificata passo dopo passo1
Recall the root test for series convergence: For a series \( \sum a_k \), if \( L = \lim_{k \to \infty} \sqrt[k]{|a_k|} \), then the series converges absolutely if \( L < 1 \), diverges if \( L > 1 \), and the test is inconclusive if \( L = 1 \).
Given \( \lim_{k \to \infty} \sqrt[k]{|a_k|} = \frac{1}{4} \), which is less than 1, the series \( \sum a_k \) converges absolutely by the root test.
Now consider the series \( \sum 10 a_k \). Since multiplying each term by a constant factor (here 10) does not affect the root limit except by a constant factor inside the root, analyze \( \sqrt[k]{|10 a_k|} \).
Note that \( \sqrt[k]{|10 a_k|} = \sqrt[k]{10} \cdot \sqrt[k]{|a_k|} \). As \( k \to \infty \), \( \sqrt[k]{10} \to 1 \), so the limit remains \( \frac{1}{4} \times 1 = \frac{1}{4} \).
Since the root limit for \( 10 a_k \) is also \( \frac{1}{4} < 1 \), the series \( \sum 10 a_k \) converges absolutely by the root test.
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Root Test for Series Convergence
The root test determines the convergence of a series by examining the limit of the k-th root of the absolute value of its terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.
Video consigliato:
Percorso guidato
07:15
Root Test
Absolute Convergence
A series ∑aₖ converges absolutely if the series of absolute values ∑|aₖ| converges. Absolute convergence guarantees convergence regardless of the signs of the terms, making it a stronger form of convergence.
Video consigliato:
Percorso guidato
07:51Choosing a Convergence Test
Effect of Multiplying Series Terms by a Constant
Multiplying each term of a series by a constant factor scales the terms but does not affect the convergence nature if the constant is finite. Specifically, if ∑aₖ converges absolutely, then ∑c·aₖ also converges absolutely for any finite constant c.
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Percorso guidato
06:45Intro to Series: Partial Sums
Pratica correlata
Domanda del libro di testo
{Use of Tech} A savings plan
James begins a savings plan in which he deposits \$100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.
To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \$100.
Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.
c.How many months are needed to reach a balance of \$5000?
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Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. When applying the Limit Comparison Test, an appropriate comparison series for ∑ (k = 1 to ∞) (k² + 2k + 1) / (k⁵ + 5k + 7) is ∑ (k = 1 to ∞) 1 / k³.
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Domanda del libro di testo
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.
Radioactive decay
A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.
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Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.
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Domanda del libro di testo
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ
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