Textbook Question
18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.
a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.
∑ (k = 0 to ∞) (–2/7)ᵏ
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.5.37a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.
Verified step by step guidance1
Step 1: Understand the given inequality and series. We have two sequences \(a_k\) and \(b_k\) such that \(0 < a_k < b_k\) for all \(k\), and the series \(\sum_{k=1}^\infty a_k\) converges.
Step 2: Recall the Comparison Test for series convergence. If \(0 \leq a_k \leq b_k\) for all \(k\) and \(\sum b_k\) converges, then \(\sum a_k\) also converges. This test gives a condition for \(a_k\) based on \(b_k\), not the other way around.
Step 3: Analyze the statement: it claims that if \(\sum a_k\) converges and \(a_k < b_k\), then \(\sum b_k\) converges. This is the reverse of the Comparison Test and is generally not true.
Step 4: To disprove the statement, consider a counterexample. For instance, let \(a_k = \frac{1}{k^2}\) which converges, and \(b_k = \frac{1}{k}\) which is larger than \(a_k\) but whose series diverges.
Step 5: Conclude that since \(\sum a_k\) converges but \(\sum b_k\) diverges in this example, the original statement is false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Comparison Test for Series
The Comparison Test helps determine the convergence of a series by comparing it to another series with known behavior. If 0 ≤ aₖ ≤ bₖ for all k and ∑ bₖ converges, then ∑ aₖ also converges. However, the converse is not necessarily true.
Recommended video:
09:25
Direct Comparison Test
Convergence of Positive Term Series
A series with positive terms converges if its partial sums approach a finite limit. Understanding that larger terms can cause divergence is key; thus, if a smaller series converges, a larger one may still diverge.
Recommended video:
06:52Convergence of an Infinite Series
Counterexamples in Series Convergence
Counterexamples demonstrate why a statement is false by providing a specific case that violates it. For series, showing a convergent smaller series and a divergent larger series disproves the claim that convergence of the smaller implies convergence of the larger.
Recommended video:
06:52Convergence of an Infinite Series
Related Practice
Textbook Question
57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.
a. Find the first four terms of the sequence of heights {hₙ}.
h₀ = 30,r = 0.25
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Textbook Question
67–70. Formulas for sequences of partial sums Consider the following infinite series.
a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.
∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]
Textbook Question
39–40. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, use Theorem 10.13 to complete the following.
a. Use Sₙ to estimate the sum of the series.
39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2
Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.
Textbook Question
57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.
a. Find the first four terms of the sequence of heights {hₙ}.
h₀ = 20,r = 0.5