Domanda del libro di testo
Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.7.5
Simplify k! / (k + 2)! for any integer k ≥ 0.
Guida verificata passo dopo passo1
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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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.
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Factorials
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.
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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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