Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces infinite sequences, providing both explicit formulas and recursion as ways to define them. Key concepts include convergence and divergence, bounded and monotonic behavior, and the use of limit laws and the Squeeze Theorem to determine limits. Graphical representation and inductive proofs further support understanding, and applications such as logistic sequences illustrate the real-world relevance of sequence analysis.

Learning Objectives

1

Define and characterize infinite sequences, including explicit and recursive definitions.

2

Explain the concepts of convergence, divergence, boundedness, and monotonicity in sequences.

3

Apply limit laws and the Squeeze Theorem to determine the limits of given sequences.

4

Utilize graphical and algebraic methods to analyze sequence behavior and verify convergence criteria.

5

Understand and model real-world phenomena with sequences, such as logistic population growth.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered list of numbers, typically defined as a function whose domain is the set of positive integers.

Infinite Sequence

A sequence with infinitely many terms, e.g., {a1, a2, a3, ...}.

Convergent Sequence

A sequence whose terms approach a fixed limit L as n becomes arbitrarily large.

Divergent Sequence

A sequence that does not converge to any finite limit, or whose terms approach infinity or oscillate indefinitely.

Bounded Sequence

A sequence is bounded if there exists a real number M such that the absolute value of every term is less than or equal to M.

Monotonic Sequence

A sequence that is uniformly increasing or decreasing. If it is also bounded, the Monotonic Sequence Theorem guarantees convergence.

Limit Laws for Sequences

Rules analogous to those for functions that allow algebraic manipulation of limits of sequences.

Squeeze Theorem for Sequences

If a sequence lies between two sequences that converge to the same limit L, then it also converges to L.

Recurrence Relation

A formula that defines subsequent terms of a sequence in terms of preceding ones, such as in the Fibonacci sequence.

Logistic Sequence

A sequence defined by a logistic difference equation used to model population growth in discrete time.

Example Problems

Example 1

(a) What is a sequence? (b) What does it mean to say that $ \lim_{n \to \infty} a_n = 8? $ (c) What does it mean to say that $ \lim_{n \to \infty} a_n = \infty? $

Example 2

(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

Example 3

List the first five terms of the sequence. $ a_n = \frac {2^n}{2n + 1} $

Example 4

.List the first five terms of the sequence. $ a_n = \frac {n^2 - 1}{n^2 + 1} $

Example 5

List the first five terms of the sequence. $ a_n = \frac {(-1)^{n-1}}{5^n} $
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Step-by-Step Explanations

QUESTION

Find limₙ→∞ (n/(n+1)).

STEP-BY-STEP ANSWER:

Step 1: Write the expression n/(n+1).
Step 2: Divide numerator and denominator by n, the highest power of n in the denominator.
Step 3: This yields (n/n) / ((n/n) + (1/n)) = 1 / (1 + 1/n).
Step 4: As n → ∞, (1/n) → 0, so the expression approaches 1/1 = 1.
Final Answer: The limit is 1.

Evaluating a Limit Using Limit Laws

QUESTION

Show that the sequence aₙ = n/(n²+1) is decreasing for n ≥ 1.

STEP-BY-STEP ANSWER:

Step 1: Write expressions for consecutive terms: aₙ and aₙ₊₁.
Step 2: Compare aₙ and aₙ₊₁ by considering the difference aₙ - aₙ₊₁.
Step 3: Use cross-multiplication to eliminate denominators and simplify the inequality.
Step 4: Verify that the resulting inequality holds for all n ≥ 1.
Final Answer: The sequence is decreasing.

Proving a Sequence is Decreasing

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Common Mistakes

  • Confusing the starting index of a sequence (e.g., starting at n=0 instead of n=1) and its impact on the definition.
  • Misapplying limit laws by not dividing both numerator and denominator by the highest power of n.
  • Neglecting the alternating nature of sequences when determining convergence, especially with sequences that involve sign changes.
  • Assuming that every bounded sequence is convergent without checking monotonicity.
  • Applying l’Hôpital’s Rule directly to sequences, even though it is defined for functions of a real variable.