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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.3.87c
Capitolo 10, Problema 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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1
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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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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Domanda del libro di testo

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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{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.


b.Find a recurrence relation that generates the sequence {Bₙ}.

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72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


c.Find a recurrence relation that generates the sequence.


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.


c. A series that converges conditionally must converge.

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


c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

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