Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.7.5

Chapter 10, Problem 10.7.5

Simplify k! / (k + 2)! for any integer k ≥ 0.

Verified step by step guidance

Recall the definition of factorial: for any integer n ≥ 0, n! = n × (n - 1) × (n - 2) × ... × 1, and 0! = 1 by convention.

Express the denominator (k + 2)! in terms of k!: write (k + 2)! as (k + 2) × (k + 1) × k!.

Rewrite the original expression \( \frac{k!}{(k + 2)!} \) by substituting the expanded form of (k + 2)!: \( \frac{k!}{(k + 2) \times (k + 1) \times k!} \).

Cancel the common factor k! in numerator and denominator, simplifying the expression to \( \frac{1}{(k + 2) \times (k + 1)} \).

Conclude that the simplified form of \( \frac{k!}{(k + 2)!} \) is \( \frac{1}{(k + 2)(k + 1)} \) for any integer k ≥ 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factorial Definition

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 up to n. By definition, 0! = 1. Understanding factorials is essential for simplifying expressions involving factorial terms.

Factorial Expansion and Cancellation

To simplify ratios of factorials like k! / (k + 2)!, expand the larger factorial to reveal common factors. For example, (k + 2)! = (k + 2)(k + 1)k!, allowing cancellation of k! in numerator and denominator.

Simplification of Rational Expressions

After canceling common factorial terms, simplify the remaining expression by performing arithmetic operations. This often reduces complex factorial ratios to simpler algebraic fractions or products.

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Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(n + 1)!⁄n!}

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21–42. Geometric series Evaluate each geometric series or state that it diverges.

31.∑ (k = 1 to ∞) 2^(–3k)

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{n³⁄(n⁴ + 1)}

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)

Textbook Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

{n¹⁰ / ln20n}

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Textbook Question

Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.