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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.1.2
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 guidance
1
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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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

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

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]

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Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (1 − n) / n2ⁿ

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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / (n³ 3ⁿ) ]

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Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

37. lim (x → 0) ln(1 + x²) / (1 - cos(x))