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 10.7.3

Chapter 10, Problem 10.7.3

Evaluate 1000!/998! without a calculator.

Verified step by step guidance

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:

\[1000 \times 999\]

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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 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.

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.

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.

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Related Practice

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∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)

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∑ (k = 1 to ∞) (3/4)ᵏ

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

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What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?

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9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

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Textbook Question

{Use of Tech} For what value of r does

∑ (k = 3 to ∞) r²ᵏ = 10?