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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.7.31c
Chapter 10, Problem 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.

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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Choosing 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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Intro to Series: Partial Sums
Related Practice
Textbook Question

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)]

Textbook Question

{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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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


c.Find a recurrence relation that generates the sequence.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

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Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.

Textbook Question

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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