Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the fundamentals of sequences, emphasizing how to list terms, derive formulas for the nth term, and determine convergence or divergence through various techniques such as L’Hôpital’s Rule and the Squeeze Theorem. An understanding of monotonic and bounded sequences provides a theoretical framework for proving convergence without necessarily knowing the limit. These tools are essential for further study in infinite series, where sequences of partial sums are used to define and evaluate sums.

Learning Objectives

1

Identify and list the terms of a sequence given either an explicit formula or a recursive definition.

2

Determine whether a sequence converges or diverges using limits, L’Hôpital’s Rule, and the Squeeze Theorem.

3

Write and interpret formulas for the nth term of a sequence based on observed patterns.

4

Apply properties of monotonic and bounded sequences to establish convergence.

5

Understand the relationship between sequences and infinite series through partial sums.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered collection of numbers defined as a function whose domain is the set of positive (or nonnegative) integers. Typically denoted as {aₙ}, where aₙ is the nth term.

Convergence

A sequence converges if, as n approaches infinity, the sequence’s terms approach a specific finite limit L.

Divergence

A sequence diverges if it does not converge to a finite limit. This may be due to oscillation or unbounded growth (or decrease).

Monotonic Sequence

A sequence that is either nondecreasing (each term is greater than or equal to the previous term) or nonincreasing (each term is less than or equal to the previous term).

Bounded Sequence

A sequence is bounded if there exists a real number that serves as an upper bound and another as a lower bound for all terms of the sequence.

Squeeze Theorem

A method used to determine the limit of a sequence by 'squeezing' it between two sequences whose limits are known and equal.

Factorial

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

Example Problems

Example 1

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=3^{n} $$

Example 2

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=\left(-\frac{2}{5}\right)^{n} $$

Example 3

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=\sin \frac{n \pi}{2} $$

Example 4

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=\frac{3 n}{n+4} $$

Example 5

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right) $$
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Step-by-Step Explanations

QUESTION

Show that the sequence with nth term aₙ = n²/(2ⁿ - 1) converges to 0.

STEP-BY-STEP ANSWER:

Step 1: Observe that as n increases, the denominator 2ⁿ grows much faster than the numerator n².
Step 2: Consider the analogous function f(x) = x²/(2ˣ - 1) for x treated as a continuous variable.
Step 3: Recognize that as x → ∞, both numerator and denominator tend to infinity, yielding an indeterminate form ∞/∞.
Step 4: Apply L’Hôpital’s Rule by differentiating the numerator to get 2x and the denominator to get 2ˣ ln2.
Step 5: Evaluate the limit limₓ→∞ [2x/(2ˣ ln2)]. Since 2ˣ increases exponentially, the limit becomes 0.
Final Answer: The sequence converges to 0.

Finding the Limit Using L’Hôpital’s Rule

QUESTION

Show that the sequence aₙ = (-1)ⁿ/(2ⁿ) converges.

STEP-BY-STEP ANSWER:

Step 1: Consider the absolute value |aₙ| = 1/(2ⁿ), which is a positive sequence that converges to 0 as n increases.
Step 2: Note that -1/(2ⁿ) ≤ (-1)ⁿ/(2ⁿ) ≤ 1/(2ⁿ) for all n.
Step 3: Since both -1/(2ⁿ) and 1/(2ⁿ) converge to 0, by the Squeeze Theorem the sequence aₙ also converges to 0.
Final Answer: The sequence converges to 0.

Applying the Squeeze Theorem

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Common Mistakes

  • Assuming convergence simply because the first few terms of a sequence appear to approach a limit without rigorous testing.
  • Confusing convergence of a sequence with convergence of an infinite series (partial sums).
  • Neglecting to verify whether a sequence is monotonic before applying convergence theorems that require monotonicity.
  • Improper application of L’Hôpital’s Rule without confirming the proper indeterminate form (such as ?/? or 0/0).
  • Overlooking the importance of checking both the upper and lower bounds when determining if a sequence is bounded.