Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ
78
views
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 10.7.3
Evaluate 1000!/998! without a calculator.
Guida verificata passo dopo passo1
Recall the definition of factorial: for any positive integer \(n\), \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
Express both factorials explicitly: \(1000! = 1000 \times 999 \times 998!\) and \$998!$ remains as is.
Rewrite the expression \(\frac{1000!}{998!}\) by substituting the expanded form of \$1000!$:
\[\frac{1000!}{998!} = \frac{1000 \times 999 \times 998!}{998!}\]
Cancel out the common \$998!$ terms in numerator and denominator, leaving:
\[1000 \times 999\]
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Factorial Definition
The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Understanding this definition is essential to simplify expressions involving factorials.
Video consigliato:
5:22
Factorials
Factorial Simplification Using Division
When dividing factorials like n! / (n - k)!, many terms cancel out because (n - k)! is a part of n!. This allows simplification to a product of k consecutive integers starting from n downwards, making calculations easier without full expansion.
Video consigliato:
5:22Factorials
Properties of Factorials in Ratios
Ratios of factorials often reduce to products of a few terms. For example, 1000! / 998! equals 1000 × 999 because 998! cancels out the terms from 1 to 998 in the numerator. Recognizing this property helps evaluate large factorial expressions efficiently.
Video consigliato:
5:22Factorials
Pratica correlata
Domanda del libro di testo
What test is advisable if a series involves a factorial term?
81
views
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)
55
views
Domanda del libro di testo
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (3/4)ᵏ
73
views
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴
41
views
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞)k² · 1.001⁻ᵏ
62
views