Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores sequences from various perspectives, including visualization techniques and recursive definitions. It explains how to generate sequences, determine their limits, and assess convergence using properties such as boundedness and monotonicity. The discussion includes detailed examples, such as recursively defined sequences and geometric sequences, to illustrate key concepts. These ideas are fundamental in mathematics and have broad applications in fields like computer science, engineering, and differential equations.

Learning Objectives

1

Understand the different methods of visualizing sequences, including plotting points on a graph and labeling points on a number line.

2

Explain and construct recursively defined sequences and recognize their patterns.

3

Analyze the convergence and divergence of sequences using limits and properties of monotonicity and boundedness.

4

Apply definitions and theorems to determine the limit of a sequence and to distinguish between bounded, convergent, and divergent sequences.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered list of numbers generated by a rule, where each number is called a term.

Recursive Definition

A method of defining a sequence where each term is expressed in terms of one or more previous terms along with initial conditions.

Convergence

A sequence converges if its terms approach a specific number (limit) as n approaches infinity.

Divergence

A sequence diverges if its terms do not approach any specific number as n increases.

Bounded Sequence

A sequence is bounded if there exist numbers K and M such that K ā‰¤ sā‚™ ā‰¤ M for all terms in the sequence.

Monotone Sequence

A sequence that is either entirely non-increasing or non-decreasing.

Factorial (n!)

The product of all positive integers from 1 up to n, with the convention that 0! = 1.

Example Problems

Example 1

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1$ $$2^{n}+1$$

Example 2

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1$ $$n+(-1)^{n}$$

Example 3

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1$ $$\frac{2 n}{2 n+1}$$

Example 4

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1$ $$(-1)^{n}\left(\frac{1}{2}\right)^{n}$$

Example 5

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1$ $$(-1)^{n+1}\left(\frac{1}{2}\right)^{n-1}$$
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Step-by-Step Explanations

QUESTION

Find the first six terms of the sequence defined recursively by sā‚™ = sā‚™ā‚‹ā‚ + 3 with sā‚ = 4.

STEP-BY-STEP ANSWER:

Step 1: Start with the initial term: sā‚ = 4.
Step 2: For n = 2, calculate sā‚‚ = sā‚ + 3 = 4 + 3 = 7.
Step 3: For n = 3, calculate sā‚ƒ = sā‚‚ + 3 = 7 + 3 = 10.
Step 4: For n = 4, calculate sā‚„ = sā‚ƒ + 3 = 10 + 3 = 13.
Step 5: For n = 5, calculate sā‚… = sā‚„ + 3 = 13 + 3 = 16.
Step 6: For n = 6, calculate sā‚† = sā‚… + 3 = 16 + 3 = 19.
Final Answer: The first six terms are 4, 7, 10, 13, 16, 19.

Recursive Sequence Example (sā‚™ = sā‚™ā‚‹ā‚ + 3, sā‚ = 4)

QUESTION

Determine whether the sequence sā‚™ = (0.8)āæ converges and find its limit.

STEP-BY-STEP ANSWER:

Step 1: Recognize that sā‚™ = (0.8)āæ is a geometric sequence with common ratio |0.8| which is less than 1.
Step 2: From the property of geometric sequences, when |x| < 1, the sequence converges to 0 as n approaches infinity.
Final Answer: The sequence converges and its limit is 0.

Convergence of a Geometric Sequence (sā‚™ = (0.8)āæ)

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Common Mistakes

  • Confusing a sequence being bounded with the sequence converging; not all bounded sequences converge, especially if they oscillate.
  • Overlooking the importance of initial conditions in recursively defined sequences, which can alter the entire sequence.
  • Assuming that a decreasing sequence always converges without checking if it is bounded below.
  • Misinterpreting the alternating signs in a sequence (e.g., 1 + (?1)?) as a sign of convergence, when in fact the oscillation can lead to divergence.