Skip to main content
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 10.4.43a
Chapter 10, Problem 10.4.43a

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


a. Find an upper bound for the remainder in terms of n.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Verified step by step guidance
1
Recognize that the series \( \sum_{k=1}^{\infty} \frac{1}{3^k} \) is a geometric series with the first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \).
Recall that for a convergent geometric series with \( |r| < 1 \), the sum to infinity is \( S = \frac{a}{1 - r} \).
The remainder \( R_n \) after summing the first \( n \) terms is the difference between the total sum and the partial sum: \( R_n = S - S_n \).
The partial sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{1 - r^n}{1 - r} \). Substitute \( a = \frac{1}{3} \) and \( r = \frac{1}{3} \) to express \( S_n \).
Express the remainder \( R_n \) in terms of \( n \) by using the formula \( R_n = S - S_n = \frac{a}{1 - r} - a \frac{1 - r^n}{1 - r} = a \frac{r^n}{1 - r} \). Substitute the values of \( a \) and \( r \) to get the upper bound for the remainder.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Convergent Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. If the absolute value of the common ratio is less than 1, the series converges to a finite sum. For example, the series ∑ (1/3)^k converges because |1/3| < 1.
Recommended video:
06:00
Geometric Series

Remainder (Error) in Infinite Series

The remainder after n terms of a convergent series is the difference between the infinite sum and the partial sum up to n terms. It measures the error when approximating the infinite sum by a finite sum. Finding an upper bound for the remainder helps estimate how close the partial sum is to the total.
Recommended video:
06:52
Convergence of an Infinite Series

Formula for Remainder in Geometric Series

For a geometric series with first term a and common ratio r (|r|<1), the remainder after n terms is given by R_n = a * r^(n+1) / (1 - r). This formula provides an explicit upper bound for the error when approximating the infinite sum by the first n terms.
Recommended video:
06:00
Geometric Series
Related Practice
Textbook Question

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

a. Find the next two terms of the sequence.

{-5, 5, -5, 5, ......}

1
views
Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of 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.

1
views
Textbook Question

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

a. Use a telescoping series argument.

1
views
Textbook Question

{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.


a.Find a recurrence relation for the sequence {dₙ} that gives the amount of drug in the blood after the nᵗʰ dose, where d₁ = 80.

1
views
Textbook Question

Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

a. For what values of p does this series converge?

Textbook Question

{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …


a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.

1
views