Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.3.31
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31.∑ (k = 1 to ∞) 2^(–3k)
Verified step by step guidance1
Identify the first term \( a \) of the geometric series by substituting \( k = 1 \) into the general term \( 2^{-3k} \). This gives \( a = 2^{-3 \times 1} = 2^{-3} \).
Determine the common ratio \( r \) by finding the ratio of the term at \( k = 2 \) to the term at \( k = 1 \). Calculate \( r = \frac{2^{-3 \times 2}}{2^{-3 \times 1}} = \frac{2^{-6}}{2^{-3}} \).
Simplify the common ratio \( r \) using the properties of exponents: \( \frac{2^{-6}}{2^{-3}} = 2^{-6 + 3} = 2^{-3} \).
Check the convergence of the series by verifying if the absolute value of the common ratio \( |r| < 1 \). Since \( r = 2^{-3} \), evaluate whether this condition holds.
If the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]. Substitute the values of \( a \) and \( r \) to express the sum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series
A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^(k), where a is the first term and r is the common ratio. Understanding this structure is essential to evaluate or determine the convergence of the series.
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Geometric Series
Convergence Criteria for Infinite Geometric Series
An infinite geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion helps decide whether the sum of infinitely many terms approaches a finite value.
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Sum Formula for Convergent Geometric Series
When an infinite geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a direct way to find the total sum without adding infinitely many terms.
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Related Practice
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
Simplify k! / (k + 2)! for any integer k ≥ 0.
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√((1 + 1 / 2n)ⁿ)}
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