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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 10.4.43c
Chapter 10, Problem 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

Verified step by step guidance
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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Key Concepts

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

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
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

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.