Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the concept of infinite sequences and series, establishing their representations and methods for evaluating convergence and divergence. It covers the epsilon definition of limits, key limit theorems, and special techniques like l’Hôpital’s Rule and the Continuous Function Theorem for sequences. Additionally, the discussion covers recursively defined sequences and the importance of bounded and monotonic sequences in determining convergence. Overall, these concepts are foundational for advancing in topics such as power series representations and applications in differential equations.

Learning Objectives

1

Explain the definition and representation of infinite sequences and series.

2

Determine convergence and divergence of sequences using the epsilon–N definition and limit theorems.

3

Apply rules such as the Sum, Difference, Constant Multiple, Product, and Quotient Rules to compute limits of sequences.

4

Use l’Hôpital’s Rule and the Continuous Function Theorem to evaluate limits when sequences are expressed in function form.

5

Analyze recursively defined and monotonic sequences to determine boundedness and convergence.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered list of numbers, often defined as a function from the set of positive integers to the reals.

Infinite Series

The sum of the terms of an infinite sequence, usually considered via partial sums.

Convergence (of a sequence)

A sequence converges to L if for every positive number ε, there exists an integer N such that for all n ≥ N, |aₙ - L| < ε.

Divergence (of a sequence)

A sequence diverges if it does not approach any single finite limit; it may increase without bound or oscillate.

Bounded Sequence

A sequence that has both an upper bound and a lower bound; that is, there exist real numbers M and m such that m ≤ aₙ ≤ M for all n.

Monotonic Sequence

A sequence that is either nondecreasing (each term is at least as large as the one before) or nonincreasing (each term is no larger than its predecessor).

Recursive Sequence

A sequence defined by one or more initial term(s) and a recursion formula that generates subsequent terms.

Factorial (n!)

The product of the first n positive integers, defined as n! = 1 × 2 × ... × n, with the convention 0! = 1.

Example Problems

Example 1

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$ $$ a_{n}=\frac{1-n}{n^{2}} $$

Example 2

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$ $$ a_{n}=\frac{1}{n !} $$

Example 3

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$ $$ a_{n}=\frac{(-1)^{n+1}}{2 n-1} $$

Example 4

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$ $$ a_{n}=2+(-1)^{n} $$

Example 5

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$ $$ a_{n}=\frac{2^{n}}{2^{n+1}} $$
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Step-by-Step Explanations

QUESTION

Show that limₙ→∞ (1/n) = 0.

STEP-BY-STEP ANSWER:

Step 1: Let ε > 0 be given.
Step 2: We need to find an integer N such that for all n ≥ N, |1/n - 0| = 1/n < ε.
Step 3: Solve 1/n < ε, which gives n > 1/ε.
Step 4: Choose N as any integer greater than 1/ε.
Final Answer: Therefore, limₙ→∞ (1/n) = 0.

Limit of 1/n

QUESTION

Show that limₙ→∞ (ln n)/n = 0 using l’Hôpital’s Rule.

STEP-BY-STEP ANSWER:

Step 1: Consider the continuous function f(x) = ln x / x for x > 0.
Step 2: As x → ∞, both ln x and x approach infinity, yielding the indeterminate form ∞/∞.
Step 3: Apply l’Hôpital’s Rule by differentiating numerator and denominator.
Step 4: The derivative of ln x is 1/x and the derivative of x is 1, so the limit becomes limₓ→∞ (1/x) = 0.
Final Answer: Hence, limₙ→∞ (ln n)/n = 0.

Limit of ln(n)/n

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Common Mistakes

  • Assuming that the convergence of the sum of two sequences implies the convergence of each individual sequence.
  • Misinterpreting the meaning of divergence to infinity as the existence of a numeric limit ‘infinity’.
  • Confusing the convergence of sequences with the convergence of infinite series, although the concepts are related but distinct.
  • Overlooking the importance of verifying that the sequence is defined for all sufficiently large n when applying limit theorems.