Textbook Question
Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.
53
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.R.15
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ
Verified step by step guidance1
Identify the given sequence: \(a_n = \frac{2^n + 5^{n+1}}{5^n}\).
Rewrite the sequence by separating the terms in the numerator over the denominator: \(a_n = \frac{2^n}{5^n} + \frac{5^{n+1}}{5^n}\).
Simplify each term: \(\frac{2^n}{5^n} = \left(\frac{2}{5}\right)^n\) and \(\frac{5^{n+1}}{5^n} = 5^{(n+1)-n} = 5\).
Express the sequence as \(a_n = \left(\frac{2}{5}\right)^n + 5\).
Analyze the limit as \(n\) approaches infinity: since \(\left(\frac{2}{5}\right)^n\) tends to 0 (because \(\frac{2}{5} < 1\)), the limit of \(a_n\) is the limit of \(0 + 5\), which is 5.
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.
Limits of Sequences
The limit of a sequence describes the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
Recommended video:
Guided course
8:22
Introduction to Sequences
Properties of Exponential Functions
Exponential functions with different bases grow at different rates. When comparing terms like 2ⁿ and 5ⁿ, the term with the larger base dominates as n increases, which helps simplify the limit by focusing on the dominant term.
Recommended video:
06:21Properties of Functions
Algebraic Manipulation of Sequences
Simplifying sequences often involves factoring and dividing numerator and denominator by the highest power term to reveal dominant behavior. This technique helps in evaluating limits by reducing complex expressions to simpler forms.
Recommended video:
Guided course
8:22Introduction to Sequences
Related Practice
Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ
48
views
Textbook Question
Geometric sums
Evaluate the geometric sums
∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.
50
views
Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.
43
views
Textbook Question
b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?
86
views
Textbook Question
Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.
43
views