Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
b.If a sequence of positive numbers converges, then the sequence is decreasing.
60
views
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.3.19a
18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.
a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.
∑ (k = 0 to ∞) (–2/7)ᵏ
Verified step by step guidance1
Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, \( a = 1 \) (since when \( k=0 \), \( (-2/7)^0 = 1 \)) and \( r = -\frac{2}{7} \).
Write the formula for the nth partial sum \( S_n \) of a geometric series:
\[ S_n = a \frac{1 - r^{n+1}}{1 - r} \]
Substitute the values of \( a \) and \( r \) into the formula to express \( S_n \) explicitly:
\[ S_n = 1 \times \frac{1 - \left(-\frac{2}{7}\right)^{n+1}}{1 - \left(-\frac{2}{7}\right)} \]
Evaluate the limit of \( S_n \) as \( n \to \infty \). Since \( |r| = \frac{2}{7} < 1 \), the term \( r^{n+1} \) approaches zero, so the limit is:
\[ \lim_{n \to \infty} S_n = \frac{1}{1 - \left(-\frac{2}{7}\right)} \]
Simplify the expression for the limit to find the sum of the infinite geometric series.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure is essential for finding partial sums and limits.
Recommended video:
06:00
Geometric Series
Partial Sum of a Geometric Series
The nth partial sum Sₙ of a geometric series is the sum of the first n+1 terms. It can be calculated using the formula Sₙ = a(1 - r^(n+1)) / (1 - r) when r ≠ 1. This formula helps in evaluating the series up to a finite number of terms.
Recommended video:
06:45Intro to Series: Partial Sums
Limit of an Infinite Geometric Series
If the absolute value of the common ratio |r| < 1, the infinite geometric series converges, and its sum is the limit of the partial sums as n approaches infinity. This limit is given by S = a / (1 - r), providing the sum of infinitely many terms.
Recommended video:
06:00Geometric Series
Related Practice
Textbook Question
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
154
views
Textbook Question
67–70. Formulas for sequences of partial sums Consider the following infinite series.
a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.
∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]
89
views
Textbook Question
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
b.Find an explicit formula for the 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.
48
views
Textbook Question
Find the first term a and the ratio r of each geometric series.
a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ
63
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.
44
views