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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 10.4.43c
Capitolo 10, Problema 10.4.43c

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

Guida verificata passo dopo passo
1
Recognize that the series given is a geometric series of the form \(\sum_{k=1}^{\infty} \frac{1}{3^k}\), where the first term \(a = \frac{1}{3}\) and the common ratio \(r = \frac{1}{3}\).
Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\). This gives the exact sum of the series.
To find the lower and upper bounds \(L_n\) and \(U_n\) for the partial sum \(S_n = \sum_{k=1}^n \frac{1}{3^k}\), calculate the partial sum using the finite geometric series formula: \(S_n = a \frac{1 - r^n}{1 - r}\).
Understand that the remainder (or error) after \(n\) terms, \(R_n = S - S_n\), can be expressed as \(R_n = \frac{a r^n}{1 - r}\). This remainder helps in estimating how close \(S_n\) is to the exact sum \(S\).
Use the partial sum \(S_n\) and the remainder \(R_n\) to establish the bounds: the lower bound \(L_n = S_n\) and the upper bound \(U_n = S_n + R_n\). This means the exact sum \(S\) lies between \(L_n\) and \(U_n\).

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Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. For |r| < 1, the infinite geometric series converges to a finite sum S = a / (1 - r), where a is the first term. Understanding this helps evaluate the exact sum of series like ∑ 1/3^k.
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Geometric Series

Partial Sums and Remainders

Partial sums are the sums of the first n terms of a series, providing an approximation to the infinite sum. The remainder (or tail) is the difference between the infinite sum and the partial sum. Estimating remainders helps find bounds on the exact value of the series.
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Integration Using Partial Fractions

Bounds on Series Sums

For convergent series, the exact sum lies between the partial sum and the partial sum plus the remainder estimate. Using remainder estimates, one can find lower and upper bounds (Lₙ and Uₙ) to bracket the true sum, ensuring an accurate approximation within a known error margin.
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Intro to Series: Partial Sums
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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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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.


c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

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Domanda del libro di testo

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.


c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

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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

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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