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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 10.3.87c
Chapter 10, Problem 10.3.87c

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.

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Recall the definition of convergence for an infinite series: A series \( \sum a_k \) converges if the sequence of its partial sums \( S_n = \sum_{k=1}^n a_k \) approaches a finite limit as \( n \to \infty \).
Consider the given series \( \sum a_k \) which converges, meaning \( \lim_{n \to \infty} S_n = L \) for some finite number \( L \).
Now examine the series \( \sum (a_k + 0.0001) \). This can be rewritten as \( \sum a_k + \sum 0.0001 \). Since \( \sum 0.0001 = 0.0001 + 0.0001 + \cdots \) is a constant term added infinitely many times, it forms a divergent series (because the partial sums grow without bound).
Because \( \sum 0.0001 \) diverges, adding it to the convergent series \( \sum a_k \) results in a series whose partial sums tend to infinity, hence \( \sum (a_k + 0.0001) \) diverges.
Therefore, the statement 'If \( \sum a_k \) converges, then \( \sum (a_k + 0.0001) \) converges' is false, and the counterexample is the constant addition of a positive number to each term.

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Key Concepts

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

Convergence of Infinite Series

An infinite series ∑ aₖ converges if the sequence of its partial sums approaches a finite limit. This means the sum of infinitely many terms settles to a specific value, rather than growing without bound or oscillating indefinitely.
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Convergence of an Infinite Series

Effect of Adding a Constant to Each Term

Adding a constant to each term of a series changes the behavior of the partial sums. Specifically, adding a nonzero constant shifts each term, which can cause the partial sums to grow without bound, potentially making the new series diverge.
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Counterexample to Test Series Convergence

A counterexample demonstrates that a general statement is false by providing a specific case where it fails. For series, showing that adding a constant to a convergent series results in divergence disproves the claim that the new series always converges.
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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

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. A series that converges conditionally must converge.

Textbook Question

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.

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.


{1, 2, 4, 8, 16, ......}

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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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