Textbook Question
Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.2
Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 1 / n!
Verified step by step guidance1
Understand the given formula for the nth term of the sequence: \(a_{n} = \frac{1}{n!}\), where \(n!\) (n factorial) is the product of all positive integers from 1 to \(n\).
Calculate \(a_1\) by substituting \(n=1\) into the formula: \(a_1 = \frac{1}{1!}\).
Calculate \(a_2\) by substituting \(n=2\) into the formula: \(a_2 = \frac{1}{2!}\).
Calculate \(a_3\) by substituting \(n=3\) into the formula: \(a_3 = \frac{1}{3!}\).
Calculate \(a_4\) by substituting \(n=4\) into the formula: \(a_4 = \frac{1}{4!}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Terms
A sequence is an ordered list of numbers defined by a specific rule or formula for its terms. Each term is identified by its position n, and the nth term aₙ gives the value at that position. Understanding how to interpret and use the formula for aₙ is essential to find specific terms.
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Introduction to Sequences
Factorial Function (n!)
The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, 0! = 1. Factorials grow rapidly and are commonly used in sequences and series.
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Evaluating Terms of a Sequence
To find specific terms like a₁, a₂, a₃, and a₄, substitute the term number n into the given formula. For aₙ = 1/n!, calculate the factorial of n and then take its reciprocal. This process allows you to generate the first few terms of the sequence explicitly.
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Related Practice
Textbook Question
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)
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Textbook Question
In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.
1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …
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Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(ln n / n)³]
Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) [3ⁿ / n³]